Stabilizing the Dual Inverted Pendulum:
A Practical Approach
by
Taylor Wallis Barton
Submitted to the Department of Electrical Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering in Electrical Engineering and Computer Science
at the Massachusetts Institute of Technology
February 2008
Copyright 2008 Taylor Wallis Barton. All rights reserved.
The author hereby grants to MIT permission to reproduce and distribute publicly
paper and electronic copies of this thesis document in whole or in part and in any
medium now known or hereafter created.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Electrical Engineering and Computer Science
February 1, 2008
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
James K. Roberge
Professor of Electrical Engineering
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Arthur C. Smith
Professor of Electrical Engineering
Chairman, Department Committee on Graduate Students
2
Stabilizing the Dual Inverted Pendulum:
A Practical Approach
by
Taylor Wallis Barton
Submitted to the Department of Electrical Engineering and Computer Science
February 1, 2008
In partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
A dual inverted pendulum system, consisting of two individual pendulums of different lengths on a
single cart, was fully designed and implemented as a demonstration of classical control theory. This
document contains an analysis of the complete control system for both a single and dual inverted
pendulum system, as well as the results of the implementation. Also presented are the all-analog
systems which were used along with an industrial permanent magnet linear synchronous motor to
control and drive the pendulum cart, including a voltage-controlled oscillator, three-phase switching
power amplifier, and acceleration feedback controller.
Thesis Supervisor: James K. Roberge
Title: Professor of Electrical Engineering
3
4
Acknowledgments
I would like to express my sincere gratitude to Professor J. K. Roberge, who has been my advisor and
mentor for the past four years. It was Professor Roberge who suggested that I attempt to implement
the dual inverted pendulum system, and this project would not have been successful without his
advice and support. I am grateful for his confidence in me and for the many opportunities he has
given me over the past years.
I would also like to thank Tracy Clark for making it possible for me to use one of MagneMotion,
Inc.’s QuickStickTM motors. This project would likely not have been such a success without this
motor at its foundation.
Thanks to Redwire, LLC. for the use of equipment and components, and to Mariano for his ideas
and suggestions, reading and editing of this document, and his constant support and encouragement.
5
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6
Contents
1 Introduction
15
2 Theory
17
2.1
2.2
2.3
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.1.1
Fast Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.1.2
Slow Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Single Pendulum Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2.1
Position Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Dual Pendulum Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.1
Position Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3.2
Maximum Phase Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3 Mechanical Design
35
3.1
Pendulum Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2
Angle Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.3
Cart Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4 Angle and Position Sensing
41
4.1
Angle Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2
Position Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.1
A Digital Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.2
An Analog Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
5 Motor Drive
47
5.1
Speed/Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.2
VCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5.2.1
Amplitude Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5.2.2
Direction Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
7
5.3
5.2.3
Offset correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
5.2.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.2.5
Three Phase Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
Power Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.3.1
Switching Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.3.2
Power Supplies for Switching Drive . . . . . . . . . . . . . . . . . . . . . . . .
57
6 Motor Control
61
6.1
Acceleration Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
6.2
Characterizing the Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
6.3
Loop Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.3.1
72
Closed Loop Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Results
7.1
7.2
7.3
75
Single Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
7.1.1
Short Pendulum Position Feedback . . . . . . . . . . . . . . . . . . . . . . . .
80
Stabilizing the Dual Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . .
83
7.2.1
Position Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
A Temporary Single Inverted Pendulum
87
B An Alternative Single Pendulum Loop
91
C Operating Instructions
95
C.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
C.2 Startup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
D List of Symbols
97
Bibliography
99
8
List of Figures
1-1 Sketches of various two-pendulum systems. The dual inverted pendulum in (a) was
implemented for this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2-1 Ideal Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2-2 This construction method for the slow pendulum results in a high effective length
while avoiding a physically large pendulum. . . . . . . . . . . . . . . . . . . . . . . .
20
2-3 Sketch of upper limit on long pendulum length (not to scale). . . . . . . . . . . . . .
22
2-4 Plot of effective length of long pendulum vs length ratio for the chosen mass ratio
and short pendulum length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2-5 Block diagram of the ideal single pendulum system. . . . . . . . . . . . . . . . . . .
24
2-6 Bode plots of ideal single pendulum system. . . . . . . . . . . . . . . . . . . . . . . .
25
2-7 Single pendulum system with position feedback. . . . . . . . . . . . . . . . . . . . . .
26
2-8 Bode and Nyquist plots of single pendulum system. . . . . . . . . . . . . . . . . . . .
27
2-9 Block diagram of single pendulum system with position feedback system and lag
compensator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2-10 Block diagram of the ideal dual pendulum system. . . . . . . . . . . . . . . . . . . .
29
2-11 Bode plots of dual pendulum major loop transfer function.
30
. . . . . . . . . . . . . .
2-12 The dual pendulum system is kept on the track with position feedback. A lag compensator is required for stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2-13 Geometric derivation of maximum phase margin based on the ideal dual pendulum
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3-1 Mechanical drawings of pendulum construction. . . . . . . . . . . . . . . . . . . . . .
36
3-2 Printed circuit board for the 2SA-10 angle sensor. The primary purpose of this board
is alignment with the pendulum axle. . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3-3 Mechanical arrangement for angle sensing. The same assembly is used for all pendulums. 37
3-4 Mechanical drawings of the pendulum cart. . . . . . . . . . . . . . . . . . . . . . . .
39
3-5 Photographs of the pendulum cart. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
9
4-1 Angle measurement circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4-2 Sketch of the location of Hall Effect sensors in the QuickStickTM stator. . . . . . . .
43
4-3 Generation of state machine input signals. . . . . . . . . . . . . . . . . . . . . . . . .
44
4-4 State machine diagram for rough position measurement. . . . . . . . . . . . . . . . .
44
4-5 The PIC outputs control which of four voltages are passed through as the position
offset, as well as the offset sign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4-6 Schematic of the analog position measurement, including a zeroing circuit activated
when the cart crosses the center of the track. . . . . . . . . . . . . . . . . . . . . . .
46
5-1 This absolute value circuit converts a velocity command into speed and direction
components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5-2 Schematic of voltage control oscillator with amplitude stabilization and direction control. 49
5-3 Root locus of (a) minor loop and (b) VCO. The minor loop sets a fixed rate of growth
for the oscillations, which are then clipped at a fixed maximum value. . . . . . . . .
49
5-4 Block diagram for VCO, shown at various levels of complexity. The minor loop can
generally be excluded from other analyses because it serves to approximate an ideal
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5-5 This circuit has either a gain of +1 or -1 depending on the switch state. . . . . . . .
52
5-6 Detailed version of the circuits represented as multipliers in Figure 5-2. . . . . . . .
53
5-7 These op amp circuits compute weighted sums of sine and cosine to create three
signals separated by 120◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-8 Switching motor drive, one phase.
54
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5-9 Waveforms demonstrating the hysteretic control. . . . . . . . . . . . . . . . . . . . .
57
5-10 Terminal voltage of a single battery with constant 2.5A discharge current. The spike
in data at around 4.25AH is due to stopping and restarting the test. . . . . . . . . .
58
5-11 Power supplies for motor drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5-12 Each phase of the motor drive is built up on a printed circuit board. . . . . . . . . .
60
6-1 A printed circuit board was built for the accelerometer. . . . . . . . . . . . . . . . .
62
6-2 Measurement configuration for results shown in Figure 6-3. . . . . . . . . . . . . . .
63
6-3 Measurements of the acceleration loop plant. . . . . . . . . . . . . . . . . . . . . . .
64
6-3 Measurements of the acceleration loop plant. . . . . . . . . . . . . . . . . . . . . . .
65
6-3 Measurements of the acceleration loop plant. . . . . . . . . . . . . . . . . . . . . . .
66
6-4 Acceleration loop compensator schematic. . . . . . . . . . . . . . . . . . . . . . . . .
68
10
6-5 Bode plot of the acceleration loop compensator, with and without the Sallen Key
filter. The Sallen Key filter is at a sufficiently high frequency such that it does not
affect the system phase margin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
6-6 Accelerometer output for the open loop and closed loop systems with a 1Hz sinusoidal
input. The closed loop system is considerably improved. . . . . . . . . . . . . . . . .
71
6-7 Final configuration for the acceleration loop . . . . . . . . . . . . . . . . . . . . . . .
72
6-8 Closed loop measurements of the acceleration loop. The loop behavior meets the
system requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
7-1 Block diagram of single pendulum system with position feedback system and lag
compensator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
7-2 Implementation of summing junction and lead compensator. . . . . . . . . . . . . . .
76
7-3 Root locus of single pendulum loop for two choices of placement for the lead compensator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
7-4 Measurement configuration for results shown in Figure 7-5. . . . . . . . . . . . . . .
77
7-5 Short pendulum system loop closed loop measurements. . . . . . . . . . . . . . . . .
78
7-6 Step response of the system shown in Figure 7-4. Top trace: measured angle output.
Bottom trace: input angle command. . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
7-7 Position measurement output when position feedback is used on the short pendulum
system without a lag compensator. This system is unstable. . . . . . . . . . . . . . .
80
7-8 Complete schematic of compensator implementation for single pendulum loop. . . .
81
7-9 Position measurement output when position feedback is connected around the dual
position system with no additional compensation. . . . . . . . . . . . . . . . . . . . .
84
7-10 Circuit implementation of compensator for dual pendulum loop. . . . . . . . . . . .
85
A-1 An adjustable slew rate limiter was used to limit the acceleration to avoid slip when
the motor was driven open loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
A-2 When the motor is driven open loop, an additional integration must be added in the
single pendulum loop compensator. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
A-3 The single inverted pendulum compensator consists of an integrator and a zero around
the natural frequency of the pendulum. . . . . . . . . . . . . . . . . . . . . . . . . .
89
B-1 Implementation of summing junction and lead compensator. . . . . . . . . . . . . . .
92
B-2 The single short pendulum system: closed loop measurement from dynamic analyzer.
93
B-3 Small signal response of short pendulum system without position feedback . . . . . .
94
C-1 Sketch of physical locations of switches on cart board. . . . . . . . . . . . . . . . . .
95
11
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12
List of Tables
6.1
Component and frequency values for acceleration loop compensator shown in Figure
6-4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
7.1
Component values for compensator circuit shown in Figure 7-2. . . . . . . . . . . . .
77
7.2
Component values for single pendulum compensator shown in Figure 7-8. . . . . . .
82
7.3
Component values for dual pendulum compensator shown in Figure 7-10. . . . . . .
84
B.1 Various iterations of component values and gain and peak overshoot measurements
for the compensator shown in Figure B-1. The small signal measurements for system
(d) are shown in Figure B-2 and Figure B-3. . . . . . . . . . . . . . . . . . . . . . . .
92
C.1 Table of switch settings for various demonstrations. . . . . . . . . . . . . . . . . . . .
96
13
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14
Chapter 1
Introduction
A system that is inherently unstable is particularly interesting to a designer of control systems
because it will not function without feedback. The dual inverted pendulum is an example of such a
system. The system plant consists of two independent pendulums constrained to fall in one plane
and controlled by the movement of a single cart. The goal is to keep both pendulums simultaneously
upright as sketched in Figure 1-1(a). This system is an extension of the single inverted pendulum [1].
It has been previously analyzed from a classical control approach in [2] and using modern control
methods in [3]. The work presented in this thesis is based on a purely classical approach to analysis
and control system design, with a focus on the practical details of implementation.
The dual inverted pendulum system should be distinguished from the double inverted pendulum,
which is a single pendulum assembly with a hinged middle as sketched in Figure 1-1(b). The double
pendulum control problem has been analyzed in the literature using modern control techniques as in
[4, 5]. Yet another variation on the inverted pendulum is the parallel-type dual inverted pendulum
analyzed in [6]. This system is sketched in 1-1(c).
(a) Dual inverted pendulum
(b) Double inverted pendulum (c) Parallel-type dual inverted pendulum
Figure 1-1: Sketches of various two-pendulum systems. The dual inverted pendulum in (a) was
implemented for this thesis.
15
The dual inverted pendulum system implemented for this thesis is stabilized using classical control
theory approaches and purely analog circuits. It is a demonstration of the sort of complicated systems
that can be stabilized using classical control. The completed system is intended to serve as a lecture
demonstration in the MIT course 6.302, Feedback Systems, a graduate level feedback course.
Chapter 2 of this document introduces the equations governing the plant, as well as the theoretical
approach to stabilization. Chapters 3 through 6 cover the various mechanical and circuit designs that
provide the foundation for implementing the pendulum system: the angle and position measurement,
the mechanical aspects of design, and the motor drive and motor control systems. Ultimately, it
is the success of these systems that determines if it is possible to stabilize the pendulums. Finally,
Chapter 7 describes the results of this implementation.
16
Chapter 2
Theory
In this chapter, the plant transfer function is derived from basic principles. Both the ideal pendulum
model and more precise models based on the pendulums built for this thesis are analyzed. Once
the plant is known, the theoretical approaches to stabilizing the single and dual inverted pendulum
systems are presented and analyzed.
2.1
Equations of Motion
The transfer function for the inverted pendulum system is found by considering an ideal inverted
pendulum consisting of a mass m at the end of a massless rod with length ℓ. Two forces act on the
mass: the downward force from gravity (Fg ) and the horizontal force from the acceleration of the
cart (Fc ) as shown in Figure 2-1. The components of these forces tangent to the circle of radius ℓ
described by the pendulum contribute to torque, and hence angular acceleration, as follows:
τg = ℓ × Fg = ℓmg sin θ
(2.1)
τc = ℓ × Fc = −ℓmẍ cos θ
(2.2)
where τg and τc are the torques from gravity and the cart acceleration, respectively, and ẍ is the
acceleration of the cart. Positive rotation is taken to be in the clockwise direction.
The angular acceleration is found from the relationship τ = J θ̈, where J is the moment of inertia
of the pendulum. For the ideal pendulum consisting of a mass on the end of a massless rod, the
moment of inertia is J = mℓ2 . The above equations thus combine to give
X
τ = J θ̈ = τg + τc = ℓmg sin θ − ℓmẍ cos θ
17
(2.3)
Fcθ
Fgθ
Fc
Fg
ℓ
θ
Figure 2-1: Ideal Inverted Pendulum
θ̈ =
ẍ
g
sin θ − cos θ
ℓ
ℓ
(2.4)
These equations are linearized about the vertical (θ ≈ 0) with the approximations for small angles,
i.e. sin θ ≈ θ and cos θ ≈ 1.
θ̈ =
g
ẍ
θ−
ℓ
ℓ
ℓθ̈ − gθ = −ẍ
(2.5)
Taking the Laplace transform of Equation 2.5 gives the ideal pendulum transfer function P (s):
P (s) =
where the time constant is τ =
−s2
−s2 /g
Θ(s)
= 2
=
X(s)
ℓs − g
(τ s + 1)(τ s − 1)
p
ℓ/g.
This expression for an ideal pendulum of course does not match the results for the pendulums
built for this thesis, particularly since the massless rod is unattainable. The constructed pendulums
have moments of inertia that are not J = mℓ2 but a more complex expression. The following
sections derive the expected equivalent length of the pendulums in terms of their physical lengths.
This analysis was used in the mechanical design of the pendulums. Once the pendulums were built,
their natural frequencies were measured by swinging each pendulum in a non-inverted position.
From this measurement an effective length can be calculated based on the ideal expression ℓ = τ 2 g.
The mechanical details are described in Section 3.1.
2.1.1
Fast Pendulum
The fast pendulum is built as a solid cylinder of steel, i.e. with the massless rod omitted. The
moment of inertia for a solid rod of length L rotated about one end is derived by integrating the
18
inertia over the length of the pendulum:
Jf =
Z
l=L
l2 dm
l=0
If the linear density of the cylinder is µ = m/L, with the incremental density µ = dm/dl, this
integral can be rewritten as
Jf =
Z
l=L
µl2 dl =
l=0
1 3 1
µL = mL2
3
3
The center of mass is L/2 from the point of rotation. With these substitutions for moment of inertia
and center of mass, Equation 2.3 is rewritten as
X
τ=
L
L
1
mL2 θ̈ = mg sin θ − mẍ cos θ
3
2
2
θ̈ =
3
(g sin θ − ẍ cos θ)
2L
Comparing this equation with Equation 2.4 reveals that the effective length must be ℓ = 23 L where L
is the physical length of the mass and ℓ is the length of an ideal pendulum having the same natural
frequency. The associated time constant for the fast pendulum is
τf =
s
ℓ
=
g
s
2L
3g
A pendulum was built from a steel cylinder with length L = 15cm as described in Section 3.1.
The natural frequency, determined by measuring its period when in a non-inverted position, indicates
an effective length of 20cm which is twice as long as expected. This discrepancy is probably due to
multiple factors: the T joint and axle contribute to the pendulum’s inertia, and the bearings and
damping from air slow the pendulum. Additionally, a longer, light plastic tube is attached around
the steel mass to give the pendulum greater visibility. This contributes to the pendulum’s inertia
but was not accounted for in the analysis.
2.1.2
Slow Pendulum
It is intuitive that having a greater ratio in lengths between the two pendulums simplifies compensation. Certainly, the system is impossible to stabilize if the two pendulums have equal lengths. The
derivation of maximum phase margin in Section 2.3.2 quantifies this intuition. From these results,
a minimum length ratio of 10 is desired, since this will give a maximum possible phase margin for
the system of 31.3 degrees.
A first attempt at a slow pendulum was made by attaching a mass (steel) to the end of a relatively
light rod (PETG tube), as described in Section 3.1. This pendulum was designed to have an effective
19
m1
ℓ1
ℓ2 m′
2
m′1
axis of rotation
m2
Figure 2-2: This construction method for the slow pendulum results in a high effective length while
avoiding a physically large pendulum.
length of 1 meter based on an expected fast pendulum length of 10cm. When measured, however,
the effective length was only 90cm, a factor of only 4.5 greater than the measured short pendulum
length. Stabilizing a system consisting of these two pendulums would nearly be impossible due to
the low phase margin. This pendulum was eventually used as a temporary pendulum for initial tests
of the system.
In order to construct a type of pendulum of this type with an effective length ten times that of
the fast pendulum, the physical length would be excessive. In addition to simply being unwieldy,
increased length leads to other problems such as flexing in the pendulum shaft. An alternative
approach is to use a pendulum as shown in Figure 2-2 to get a longer effective length without having
a physically awkward construction [7]. If the masses and lengths of the two opposing pendulums
shown are equal, then the object will be balanced and will not exhibit a pendulum-like behavior.
By incrementally increasing m1 or ℓ1 , this mechanism will behave as an inverted pendulum with a
very slow time constant. In this figure, m1 and m2 are steel masses at the end of rods of length ℓ1
and ℓ2 , which have mass m′1 and m′2 respectively.
If the masses of the rods are ignored, the total torque acting on this pendulum is
Στg + Στc = ℓ1 m1 g sin θ − ℓ2 m2 g sin θ − ℓ1 m1 ẍ cos θ + ℓ2 m2 ẍ cos θ
= (ℓ1 m1 − ℓ2 m2 )(g sin θ − ẍ cos θ)
and the total rotational inertia is simply the sum of the inertias of the two masses at their respective
lengths:
J = m1 ℓ21 + m2 ℓ22
Making the appropriate substitutions in the analysis from Section (2.1) above gives an equation of
motion
θ̈ =
ℓ 1 m1 − ℓ 2 m2
m1 ℓ21 + m2 ℓ22
20
(g sin θ − ẍ cos θ)
The expression for the effective length of this structure is
ℓs =
m1 ℓ21 + m2 ℓ22
ℓ 1 m1 − ℓ 2 m2
and the time constant associated with the pendulum is
p
τs = ℓ/g =
s
m1 ℓ21 + m2 ℓ22
g(ℓ1 m1 − ℓ2 m2 )
These expressions can be generalized for design purposes. If m1 is chosen to be some multiple of
m2 , i.e. m1 = km m1 = km m, the effective length can be rewritten as
ℓs =
km ℓ21 + ℓ22
km mℓ21 + mℓ22
=
ℓ1 km m − ℓ2 m
km ℓ1 − ℓ2
Similarly, choosing ℓ1 = kℓ ℓ2 = kℓ L gives
ℓs = L
km kℓ2 + 1
km kℓ − 1
where L is the physical length of the noninverted pendulum and m is the associated mass.
This result is elegant and – as with the ideal pendulum – does not depend on the absolute
values of the masses, only their ratio. However, from experience the mass of the rods should not be
neglected. When m′1 and m′2 are included, the torque equation becomes
1
1
m1 ℓ21 + m′1 ℓ21 + m2 ℓ22 + m′2 ℓ22 =
3
3
ℓ2
ℓ1
ℓ1 m1 + m′1 g sin θ − ℓ2 m2 + m′2 g sin θ
2
2
ℓ1 ′
ℓ2
− ℓ1 m1 + m1 ẍ cos θ + ℓ2 m2 + m′2 ẍ cos θ
2
2
The above analysis is repeated on this equation, with m1 = km m2 = km m and ℓ1 = kℓ ℓ2 = kℓ L. The
rod masses are proportional to the lengths as m′1 = ρr ℓ1 and m′2 = ρr ℓ2 . Likewise, the relationship
between the density by length of steel and that of the plastic rod is written as ρm = kρ ρr . The
length of the steel mass m2 is assumed to be 5cm. When the mass of the rods is included, the
expression for effective length becomes
ℓ′s =
0.05kρ L2 (km kℓ2 + 1) + 13 L3 (kℓ3 + 1)
0.05kρ L(km kℓ − 1) + L2 (kℓ2 − 1)
Clearly the pendulum length can be chosen to be arbitrarily large by setting the denominator
term to zero. However, there is a lower limit to natural frequency due to the finite length of the
track. Assume the cart is at one end of the track and the pendulum leans in with the maximum
allowed angle, approximately 5 degrees. The control system will be set up to accelerate the cart so
21
that the pendulum “sees” gravity as exerting a force in the opposite direction from actual. If this
is the case, the distance x the cart must travel to bring the pendulum angle to zero is (ℓs sin θ),
where ℓs is the effective length of the pendulum, as sketched in Figure 2-3. The maximum allowable
pendulum length is therefore
ℓs,max =
0.5m
= 5.7m
sin(5◦ )
A plot of ℓ′s vs kℓ is shown in Figure 2-4, for L = 15cm and for km = 0.75. The ratio of masses
is chosen to be less than one to allow for a greater difference in the lengths. The ratio of densities is
derived from the physical dimensions and from typical material densities found in [8]. It is clear from
this plot that small changes in length have a large effect on the effective length of the pendulum.
Furthermore, even the more precise expression for ℓ′s derived above neglects the inertia of the T and
elbow joints used to connect the pendulum to its axle. With this in mind, the slow pendulum was
designed with mass m1 initially moveable. By incrementing the length ℓ1 until the effective length
fell within the allowed range shown in Figure 2-4, an effective length of 2.65 meters was measured.
The mass was then fixed in this position.
x
ℓs
θ
x=0
Figure 2-3: Sketch of upper limit on long pendulum length (not to scale).
22
12
11
Mass ratio km = 0.75
10
Short pendulum length L = 15cm
Effective length ℓ′s
9
8
7
maximum ℓ′s
6
5
4
3
minimum ℓ′s
2
1
0
1.1
1.2
1.3
1.4
Length ratio kℓ
Figure 2-4: Plot of effective length of long pendulum vs length ratio for the chosen mass ratio and
short pendulum length.
23
2.2
Single Pendulum Control
The single pendulum control is based on the approach developed by Roberge in his undergraduate
thesis [1] and is intuitive to anyone who has attempted to balance a yardstick in one hand. An angle
from vertical results in a force on the pendulum, which must be corrected by an acceleration by the
hand in the direction the yardstick is falling. If the pendulum base is accelerated at e.g. twice the
rate of fall of the top of the pendulum, the pendulum will move towards vertical. The mechanical
system takes this approach, accelerating the cart at some multiple of the pendulum angle.
A block diagram of the ideal single pendulum system is shown in Figure 2-5. As shown, the
system consists of the pendulum, the motor, and some compensator G(s). For this thesis the
motor
G(s)
+
Km
s2
x
+
θ
−s2 /g
(τp s+1)(τp s−1)
pendulum
Figure 2-5: Block diagram of the ideal single pendulum system.
motor was controlled with an acceleration control loop, so the ideal transfer function of the motor
block is Km /s2 . The implemented control system is described in Chapter 6. If G(s) = 1, the
loop transfer function has characteristics as shown in Figure 2-6(a). Because the uncompensated
system is symmetric in the complex plane about the imaginary axis, the phase shift is always 180
degrees. When G(s) is a lead network, the system becomes stable as shown in Figure 2-6(b).
Theoretically, lead frequency and loop gain can be increased without bound for a high crossover
frequency. In practice, however, higher frequency poles begin to introduce negative phase shift that
limits crossover.
2.2.1
Position Feedback
One problem with this simple approach is a result of measurement error. If there is an offset
measurement in the angle – which is inevitable in a real system – the system will drive the pendulum
to a nonzero angle. This angle can be maintained with a constant cart acceleration, but the cart will
quickly leave the track. Feedback must therefore be used to control the cart position. The position
of the cart is measured and added as an offset to the angle measurement, as shown in Figure 2-7(a).
The sign is chosen to result in positive feedback around the motor. As the cart moves to the end of
the track, the position command leans the pendulum towards the center, causing the cart to move
in that direction.
24
101
10−1
6
180
L (degrees)
|L(jω)|
100
10−2
10−3 −1
10
100
101
102
ω (rps)
(a) Uncompensated system
10
1
−90
−135
6
10−1
L (degrees)
|L(jω)|
100
10−2
−180
10−3 −1
10
100
101
ωc
102
ω (rps)
(b) Compensated with lead network
Figure 2-6: Bode plots of ideal single pendulum system.
25
kp
+
jω
motor
G(s)
Km
s2
x
≈
+
+
θ
σ
−s2 /g
(τp s+1)(τp s−1)
pendulum
(a) Block diagram
(b) Plot of open loop pole and zero locations
Figure 2-7: Single pendulum system with position feedback.
This simple approach to position feedback does not result in a stable system. In root locus
terms, the position feedback moves the two motor poles from the origin into the right and left half
planes. The resulting pole-zero plot is shown in Figure 2-7(b). Stability can best be determined
through Nyquist plots, which must show two encirclements of the −1 point since the open-loop
transfer function has two poles in the right half plane. The plots in 2-8(a) and (b) indicate that
for stability the phase must be more negative than −180 degrees at the low-frequency crossover, so
that the Nyquist curve crosses the negative real axis. One way to achieve this negative phase shift
is with air damping. Damping moves one of the zeros in the pendulum transfer function off of the
origin and into the left half plane [1]. Now, this zero contributes a negative 90 degree phase shift at
DC. For a sufficiently large damping coefficient, the angle will remain below −180 degrees when the
magnitude is one. Plots of a damped system are shown in Figures 2-8(c) and (d).
If the air damping coefficient is insufficient, an electrical equivalent can be used. By placing a
lag network around the lower crossover frequency, the phase shift can be brought more negative
than −180 degrees. The lag network Glag (s) in the resulting block diagram shown in Figure 2-9 has
transfer function
Glag (s) =
τs + 1
ατ s + 1
Figures 2-8(e) and (f) demonstrate this system to be stable.
26
101
−90
−135
−1
6
10−1
L (degrees)
|L(jω)|
100
10−2
−180
10−3 −2
10
100
10−1
101
ω (rps)
102
103
104
(a) Bode plot of system with position feedback
101
−90
0
−135
|L(jω)|
10−1
−180
10−2
STABLE
−1
6
10−3
L (degrees)
10
(b) This system is unstable since n 6= 2
−225
10−4
10−5 −3
10
10−2
10−1
100
101
ω (rps)
102
103
−270
104
(c) If some damping is present, the system is stable with
position feedback
101
(d) The damped system is stable in the region
indicated
−135
10−1
−180
10−2
−225
STABLE
−1
6
|L(jω)|
100
L (degrees)
−90
10−3 −2
10
10−1
100
101
ω (rps)
102
−270
103
(e) A lag network has the same effect as damping
(f) The compensated system is stable in the region indicated
Figure 2-8: Bode and Nyquist plots of single pendulum system.
27
kp
+
motor
Km
s2
G(s)
+
x
+
τ s+1
ατ s+1
−s2 /g
(τp s+1)(τp s−1)
Glag (s)
pendulum
Figure 2-9: Block diagram of single pendulum system with position feedback system and lag compensator.
2.3
Dual Pendulum Control
It is evident that the two pendulums controlled must have different effective lengths. If they are
the same length, then any acceleration of the cart would affect their angles equally so the difference
between their angles would be constant. The intuitive way to stabilize two pendulums of different
lengths, then, is to “catch” the shorter one first, since it falls more quickly, and to try to do so in
a way that brings the taller one closer to vertical. The strategy is to drive the cart such that the
short pendulum’s angle is always, say, twice that of the slow pendulum, so that the next cart move
will bring both of them to upright.
This intuitive approach describes a minor loop configuration, where the fast pendulum is set up
as in the single pendulum case to take an angle command, and is driven as a function of the slow
pendulum’s angle. An initial block diagram is shown in Figure 2-10. The minor loop is identical to
the single pendulum system without position feedback shown in Figure 2-5 and is compensated in
the same way. If the minor loop crossover frequency and loop gain can be made sufficiently high,
its closed loop transfer can be approximated as
−(τf s + 1)(τf s − 1)
X
(s) =
Θ
s2 /g
When Gs (s) = 1, the loop transmission of the major loop is therefore
L(s) =
s2 /g
(τf s + 1)(τf s − 1)
(τf s + 1)(τf s − 1)
=
2
(τs s + 1)(τs s − 1)
s /g
(τs s + 1)(τs s − 1)
(2.6)
This transfer function is plotted in Figure 2-11(a). Between 1/τs and 1/τf , the magnitude drops off
as −40dB/decade, but the phase is always equal to −180 degrees because the system is symmetrical
in the complex plane about the imaginary axis. In order to stabilize this system, a lead compensator
28
motor
Gs (s)
X
+
Gf (s)
+
Θf
Km
s2
−s2 /g
(τf s+1)(τf s−1)
fast pendulum
Θs
−s2 /g
(τs s+1)(τs s−1)
slow pendulum
Figure 2-10: Block diagram of the ideal dual pendulum system.
is added with the form
Gs (s) =
τs s + 1
τf s + 1
(2.7)
This approach is similar to the one taken to stabilize the single pendulum, only in this case the
location of the compensator pole has an upper limit equal to the natural frequency of the fast
pendulum. If the pole is placed at a higher frequency, the magnitude will begin to increase because
of the two zeros from the fast pendulum transfer function, lowering the gain margin. A Bode plot
of the lead-compensated system is shown in Figure 2-11(b).
29
100
−180
6
10−1
L (degrees)
|L(jω)|
−135
10
−225
−2
10−2
10−1
100
ω (rps)
101
102
(a) Uncompensated system
10
1
−135
L (degrees)
100
6
|L(jω)|
−90
φM
10
−180
−1
10−2
10−1
100
ω (rps)
ωc
101
102
(b) Compensated with lead network
Figure 2-11: Bode plots of dual pendulum major loop transfer function.
30
2.3.1
Position Feedback
As with the single pendulum system, angle measurement offset makes this ideal analysis insufficient.
To keep the cart on the track, the cart position is measured and added as an offset to the angle
measurement as shown in Figure 2-12(a). Note that for this system, there is no position feedback in
the minor loop, as the major loop position feedback corrects for any offsets in the minor loop. As
in the single pendulum case, the effect of the position feedback is to move the two poles from the
minor loop transfer function off of the origin and into the left and right half planes. The resulting
system can again be compensated using a lag network, as shown in Figure 2-12(b) and (c). Because
the open-loop system has two poles in the right half plane, the Nyquist plot demonstrates stability
when there are two negative encirclements of the −1 point.
kp
motor
X
Km
s2
Gf (s)
Gs (s)
Θf
−s2 /g
(τf s+1)(τf s−1)
fast pendulum
τ s+1
ατ s+1
−s2 /g
(τs s+1)(τs s−1)
lag
slow pendulum
(a) Block diagram of dual pendulum system with position feedback and lag
compensator
101
10−1
−180
10−2
−225
10−3
−270
6
−135
|L(jω)|
100
L (degrees)
−90
STABLE
−1
10−4
10−5 −3
10
10−2
10−1
100
101
ω (rps)
102
103
104
(b) Bode plot of compensated system
(c) Nyquist plot demonstrating system stability
Figure 2-12: The dual pendulum system is kept on the track with position feedback. A lag compensator is required for stability.
31
2.3.2
Maximum Phase Margin
Equations 2.6 and 2.7 lead to a relationship between the length ratio of the two pendulums and the
maximum attainable phase margin. Assuming the loop compensator is a lead network as above, the
maximum phase margin is set by the natural frequencies of the pendulums. The simplified ideal
loop transfer function is
L(s) =
(τf s + 1)(τf s − 1) (τs s + 1)
τf s − 1
=
(τs s + 1)(τs s − 1) (τf s + 1)
τs s − 1
as plotted in the complex plane in Figure 2-13.
The peak phase shift from this system occurs at the geometric mean of the two frequency el√
ements, i.e. at ωc = ωf ωs . The net phase shift is equal to the difference between the phase
contributed by the zero (θz ) and that contributed by the pole (θp ). This difference is equal to the
angle labeled φM in Figure 2-13 by geometric construction. The trigonometric law of sines states
that in a triangle, the ratios of each angle to its opposite leg are equal. Writing this relationship
results in the equation
sin φM
sin(180 − θz )
= p
ωf − ωs
ωs (ωs + ωf )
p
√
ωf ωs / ωf (ωf + ωs )
p
=
ωs (ωs + ωf )
1
=
ωf + ωs
ωf − ωs
φM = arcsin
ωf + ωs
This equation can be rewritten with the substitution ω = 1/τ =
p
g/L to relate the phase margin
to the length ratio of the pendulums. For the length ratio of 13.3 used in this implementation, the
maximum ideal phase margin is
φM = arcsin
!
p
Ls /Lf − 1
p
= 34.7◦
Ls /Lf + 1
A real-world system will have higher order poles that will have additional negative phase contributions at crossover. It is therefore very important when implementing this system to minimize this
phase shift.
32
jω
q
ωs (ωs + ωf )
q
φM
√
ωf (ωf + ωs )
ωf ωs
θz
θp
ωs
ωf − ωs
σ
Figure 2-13: Geometric derivation of maximum phase margin based on the ideal dual pendulum
system.
33
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34
Chapter 3
Mechanical Design
Although the nominal topics of this thesis are feedback control and circuit design, a large portion
of the work involved mechanical design and construction. This section describes the construction of
each piece. Initial mechanical design was performed with the SolidWorksTM CAD Software Package.
R Center water jet in the
Where practical, flat pieces were cut out using the OMAX JetMachining
Hobby Shop at MIT. In critical areas near the permanent magnets and magnetic angle sensors,
ferromagnetic materials were avoided.
3.1
Pendulum Construction
Three pendulums were built for this thesis: a medium-length (nominally 1 meter) temporary pendulum for initial testing, a fast (nominally 10cm) pendulum, and a slow pendulum (nominally 2.6
meters). All of the pendulums are constructed from the same basic materials. The axles are turned
out of PET rod and press fit into delrin ball bearings with glass balls. These materials were chosen
because the axle houses the magnetic angle sensor, so steel could not be used. A PVC T pipe fitting
attaches the pendulum to its axis. This T is bolted to the axle to prevent rotation. The pendulums
are constructed out of clear PETG hollow tube which fits into the T joint. Since this is not an exact
fit, the gap is filled with epoxy. Each of the pendulums is sketched in Figure 3-1.
The temporary pendulum is meant to approximate an ideal pendulum consisting of a mass at
the end of a massless rod. Thus a 3.2cm steel cylinder is attached inside the end of the relatively
massless hollow tube. The pendulum is stiffened by epoxying pieces of 3/16” diameter PETG
rod inside the hollow rod across its diameter. The fast pendulum is constructed in a similar way
to the temporary pendulum, except the steel pendulum is attached at the bottom of the PETG
tube. The tube is extended above the steel to give the pendulum greater visibility for e.g. a lecture
demonstration. The slow pendulum is constructed using the technique introduced in Section 2.1.2 to
35
steel mass
stiffening
rods
hollow PETG rod
T joint
bearings
pillow block
(a) Temporary pendulum, end view
(b) Temporary pendulum, side view
hollow PETG rod
mp
T joint
steel pendulum
pillow block
bearings
(c) Short pendulum, end view
(d) Short pendulum, side view
steel mass
elbow joint
elbow joint
T joint
block for angle
measurement
pillow block
steel mass
(f) Long pendulum, side view
(e) Long pendulum, end view
Figure 3-1: Mechanical drawings of pendulum construction.
36
avoid an unwieldy length. By adding a pendulum in the noninverted position, the natural frequency
is increased. This counterweight is attached with an elbow joint as shown in Figure 3-1. For the
slow pendulum, both the T and elbow joints are threaded so that the two pendulum parts can be
removed by unscrewing. This allows the slow pendulum to be removed for a demonstration of the
single pendulum compensation loop.
3.2
Angle Measurement
The angle measurement is made using the Sentron 2SA-10 magnetic sensing IC as described in
Section 4.1. For this measurement, a magnet was inserted into a bored-out space in the pendulum
axle. A printed circuit board was manufactured for the 2SA-10 sensor, with spring pins in matching
holes in the board and pillow blocks to hold the board in alignment. The layout is shown in Figure
3-2. The two larger holes at the center sides hold the spring pins, and the three smaller holes match
with tapped holes for screws. The mechanical setup with the pendulum axle and pillow block is
shown in Figure 3-3.
Figure 3-2: Printed circuit board for the 2SA-10 angle sensor. The primary purpose of this board
is alignment with the pendulum axle.
Figure 3-3: Mechanical arrangement for angle sensing. The same assembly is used for all pendulums.
37
3.3
Cart Design
The motor at the heart of this thesis is a QuickStickTM Linear Synchronous Motor (LSM) lent for
this thesis by MagneMotion, Inc. The QuickStickTM is made up of a one meter long rectangular
stator and an array of permanent magnets. When the stator windings are energized, the resulting
magnetic field interacts with the permanent magnets on the rotor to produce thrust. The motor
drive designed for this system is discussed in detail in Chapter 5. In order for this drive to be useful,
the permanent magnet array must be affixed to a cart that can move freely along the length of the
motor.
The pendulum cart designed for this thesis is a wheeled, U-shaped cart that fits loosely over the
stator. This design was chosen to avoid constructing an alternative rail-type system because of the
perceived difficulty of constructing a sufficiently straight track parallel to the motor. The U-shape
is maintained by brackets that hold the aluminum sides and cart base together. The magnet array
is sandwiched below the base of the cart with a piece of 1/8” delrin which in addition to a 1/16”
clearance between the delrin and stator sets the motor gap. Mounting holes in the cart sides hold
four rubber wheels and four inward-facing ball casters. Because of the spacing between the casters
and stator, the casters do not typically touch the stator, so the typical frictional force is reduced.
This design is based on the idea that a restorative force will keep the magnets aligned with the
stator. In practice, however, the cart will sometimes drift to the side and the casters will come
in contact with the stator. Figures 3-4 and 3-5 show mechanical drawings and photographs of the
design.
The pendulum cart base is a 5.75 inch by 10 inch solid piece of aluminum which serves as a
mounting point for the various hardware on the cart. In addition to the two pendulums’ pillow
blocks described above, the pendulum cart carries all of the circuits described in this document with
the exception of the voltage-controlled oscillator and the power electronics for the motor drive. As
a result, the single connection between the pendulum cart and the other circuitry is a RG-174 coax
cable carrying the velocity command signal. Four nine-Volt batteries are mounted on the cart to
power the control system. The nominal ±18V is regulated down to ±12V and +5V with LM7812,
LM7912, and LM7805 linear regulators. The cart base is electrically grounded to provide some
amount of shielding between the cart circuitry and the magnetic field from the stator windings.
38
Figure 3-4: Mechanical drawings of the pendulum cart.
39
Figure 3-5: Photographs of the pendulum cart.
40
Chapter 4
Angle and Position Sensing
For an ideal system, only the angle of the pendulum(s) must be known to close a control loop. By
measuring the angle with respect to vertical, a loop can be designed to drive the pendulum upright.
Due to the inherent offsets in real-world angle measurements, however, the pendulum will not be
precisely balanced when the angle measurement reads zero. To maintain this non-zero pendulum
angle, the cart must move with some constant acceleration to offset the force of gravity. Without
a position control loop, the cart will run off the end of the track. The position of the cart must
therefore be measured in addition to pendulum angle so that two feedback loops can be used. This
chapter describes how these two measurements are made.
4.1
Angle Sensing
Both the fast and slow pendulum have an identical arrangement for measuring angle displacement.
The angle is measured with Sentron’s 2SA-10 two axis magnetic sensing IC [9]. The 2SA-10 uses
four sets of Hall Effect sensors to measure x- and y-components of the magnetic field parallel to its
surface. When paired with a magnet polarized across its diameter, it can detect absolute angular
position. The main advantage of the 2SA-10 is that it is a contactless measurement. Delrin bearings
with glass balls are used so as not to disrupt the measurement magnetic field (see Chapter 3 for
mechanical details). Interfering fields from the permanent magnet array and stator windings of the
motor are shielded by a steel plate built into the back of the magnet array.
The outputs of the 2SA-10 are X_Out, the x-component of the measured magnetic field; Y_Out,
the y-component; and CO_Out, the reference voltage corresponding to zero magnetic field. The
magnet is placed close to the sensor so that the full-scale field exceeds the 45mT range of the 2SA10 and the device saturates at peaks in the field. With the magnet oriented so that either the X or
Y field is close to zero when the pendulum is vertical, the signal is increased in the range of angles
41
+5V
R1
R2
+5V
100nF
2SA-10
R2
Cos(θ)
Ref
Sin(θ)
AD524
(x34)
angle
R1
Figure 4-1: A single output of the 2SA-10 is used. By comparing it to the common output and
amplifying the signal, an angle signal is generated that is centered at zero and is full scale when the
angle corresponds to the maximum angle that can be stabilized.
where the pendulum can be stabilized, and does not saturate until the pendulum is at least 30◦ off
vertical. The 2SA-10 has a typical bandwidth of 0-15kHz so it does not contribute appreciable phase
shift at the frequencies relevant to this system.
As shown in Figure 4-1, the output of the 2SA-10 is amplified using a precision instrumentation
amplifier configured to have a gain of 34. Since the 2SA-10 is operated from a single supply, the
output must be offset by CO_Out to shift the signals so that a zero angle corresponds to zero Volts.
The reference CO_Out is typically at 2.5V when the 2SA-10 is operated from a single 5V supply.
Since it is difficult to align the magnet perfectly within the pendulum axle, potentiometers are
used to trim the reference voltage so that the output of the amplifier is as close to zero as possible
when the pendulum is vertical. Any further offset in the angle measurement is taken care of by the
position feedback in the control system. The gain of the angle measurement was measured to be
1.6V/degree = 91V/radian.
42
4.2
Position Sensing
Two methods of measuring linear position were explored for this thesis. First, a digital approach
using a microcontroller was attempted, then an analog approach of integrating the velocity signal.
4.2.1
A Digital Approach
The MagneMotion QuickStickTM has built-in Hall Effect sensor pairs that measure the magnetic
field from the magnet array at eight points along the motor. The outputs from the Hall Effect
sensors do not include the field from the stator windings (signal is only present when the magnet
array is above the Hall Effect sensors), so the motor drive does not affect these signals. Figure 4-2
shows how the motor can be broken up into eight segments based on when each Hall Effect sensor
pair is active. Note that this is a simplistic view as there is in fact overlap between neighboring
pairs due to the length of the magnet array used.
d
b
D
C
c
f
B
A
a
E
F
e
h
G
H
g
HES sensors
motor
HES sensors
Figure 4-2: The motor can be arranged in eight sections based on which Hall Effect sensors are
active. Lowercase letters indicate the range over which each Hall Effect pair has a non-zero output.
Each of the lower case letters in Figure 4-2 represents the range over which a pair of Hall Effect
sensors has a non-zero output. As the cart moves over a sensor pair, the two sensors output signals
in quadrature as shown in Figure 4-3. By comparing these two outputs, a signal is generated that
switches as the magnet array moves across it, and is constant when the magnet array is absent.
Hysteresis prevents the comparator from switching when the sensor outputs are nominally equal.
The switching signals are then used as the input to a state machine, with rising or falling edges
triggering events.
The state machine shown in Figure 4-4 sets the position output to one of eight values depending
on which of the eight sections the cart is over. It assumes the cart begins in the center of the track,
i.e. in the state A/E with a corresponding position output of zero. If the cart moves to the left, the
active Hall Effect sensors are first only pair a, then pairs a and b. When b becomes active, the state
machine transitions to state B and the position output is stepped up to a nonzero value. As the
cart continues to the left, there will be a rising or falling edge from sensor pair c that triggers the
machine to switch to state C. When in this state, as long as only b or c has rising or falling edges
the state remains the same and the position output is set to a constant value. If the cart returns to
43
motor
region
B
C
A
cart over pair b
pair b
sensor
outputs
cart over pair a
pair a
sensor
outputs
comparator
b output
event b
comparator
a output
event a
Figure 4-3: As the cart moves over the sensor pairs they output quadrature signals corresponding to
the magnetic field from the permanent magnet array. Comparing the two outputs gives a switching
signal which is used as an input to a state machine.
the right, sensor pair a will switch and the state will be set to B. Finally, if the cart returns to the
center the state does not switch back to A/E until the e pair is active.
A PIC16F628A is used to implement this state machine. The output of the PIC is a three-bit
indicator corresponding to the state of the state machine. The two least significant bits control a 4:1
analog switch which passes through the voltage levels corresponding to the different distances from
the center. The MSB controls the sign of the output using a gain block with ±1 gain described in
detail in Section 5.2.2. This circuit is shown in Figure 4-5.
As the cart moves within one of the eight sections of the track, the position offset value will
c
b
b
D
a
C
d
d
a
e
B
c
c
f
A/E
b
b
e
a
F
G
e
f
H
f
g
Figure 4-4: State machine diagram for rough position measurement.
44
g
h
g
a
e
f
h
R
R
v1
v2
v3
v4
−
R1
bit1
bit0
+
vpos
C1
R
bit2
Figure 4-5: The PIC outputs control which of four voltages are passed through as the position offset,
as well as the offset sign.
remain constant even though the position is changing by up to 1/8 meter. This means that the
effective voltage/position gain will vary. That is, the gain will have a step increase when the state
machine changes state, then it will linearly decrease until the next transition. These changes in gain
are acceptable for use in a feedback loop so long as the system is stable for both extremes. There is,
however, a potential problem because the position measurement is added as an offset to the angle
measurement. The steps in the position signal are interpreted as a step in pendulum angle by the
system, which is not necessarily equipped to handle such an input. Therefore for the single long
pendulum a low-pass filter is added at the output of this position measurement to smooth over the
transitions. Although this is sufficient for the single long pendulum case, for the short pendulum
and for the dual pendulum system it is necessary to reduce the size of the position signal steps by
using a more accurate position measurement. This system is described in the next section.
4.2.2
An Analog Approach
A position measurement can instead be derived by analog integration of the velocity command. This
approach relies on two assumptions: first, that the velocity command is followed accurately by the
cart, and second, that offsets do not cause the integration to drift over time. The first assumption is
satisfied by the motor control loop described in Chapter 6. The integrator drift problem is avoided
by resetting the position measurement to zero when the cart crosses the center of the track. Finally,
the integrator is designed with a sufficiently low constant of integration such that the integrator will
not saturate over the full length of the track.
The reset command works by shorting across the integrating capacitor with an analog switch
(AD7510). A center crossing is detected when a brush on the cart contacts a piece of copper foil
on the floor of the track near the center. The copper foil is placed close to the stator so that the
wheels do not run over it. When not in contact with the foil, this signal is held at ground with a
pull-down resistor as shown in Figure 4-6. A Schottkey diode clamp and series resistance provide
input protection from voltage spikes due to poor commutation. The AD7510 outputs are protected
45
+15V
1/4 AD7510
470Ω
+5V
13kΩ
330Ω
2µF
velocity
−
100kΩ
LF356
position
+
Figure 4-6: Schematic of the analog position measurement, including a zeroing circuit activated
when the cart crosses the center of the track.
with a series resistor to prevent excessive currents. This circuit was built and measured to have an
output scale factor of 18.4V/meter.
46
Chapter 5
Motor Drive
position
feedback
Gs (s)
Gf (s)
VCO
motor/ ẍ
cart
motor
drive
1/s2
x
Ga (s)
θf
fast
pendulum
θs
Glag (s)
slow
pendulum
The QuickStickTM LSM used for this thesis is a three-phase synchronous linear motor with a
permanent magnet rotor. Although MagneMotion also provided an associated control system and
drive, for educational purposes a complete motor drive was designed and built. The motor drive
described in this chapter takes a velocity command and converts it into a three-phase drive to
appropriately move the rotor. The three-phase signals are generated by a VCO. Then a switching
power amplifier for each phase drives the motor windings to move the pendulum cart.
5.1
Speed/Direction
The velocity command input must be converted to speed and direction signals before going to the
VCO for reasons described in the next section. The precision rectifier circuit used, a variation of
one found in [10], is shown in Figure 5-1. The output of the circuit is vO = −vI − 2vA , where
vA = 0 when the input is negative, and vA = −vI when the input is positive. For negative inputs,
the transistor conducts, which both pulls vA to zero and provides the direction signal through R2 .
It is also necessary to include a series diode because the reverse breakdown of the transistor is lower
than the maximum signal swing.
47
+5V
R2
direction
R
−
+
R
vi
R/2
R
vA
−
+
vo
R
Figure 5-1: This absolute value circuit converts a velocity command into speed and direction components.
5.2
VCO
The speed and direction signals drive the quadrature voltage controlled oscillator (VCO) shown
in Figure 5-2. The VCO generates two sinusoids of varying frequency that are 90 degrees out of
phase. These signals, combined in the correct proportions, produce three sinusoids that are 120
degrees out of phase, which are then used to command the motor drive. The VCO in basic form is
a pair of integrators with one inversion. By scaling the inputs to the two integrators by the same
factor, the natural frequency can be changed to create an output frequency proportional to input
voltage. The actual VCO is somewhat more complicated than this. Figure 5-4 shows a series of
block diagrams breaking down the system into its various parts, which are described in detail in the
following sections.
5.2.1
Amplitude Stabilization
In the ideal case the two-integrator system would have a root locus with two poles starting at the
origin and going up and down the imaginary axis. As long as there is no real component to the
poles, the oscillations will not grow or shrink in magnitude. In practice, however, nonidealities such
as capacitor ESR will place the poles off the origin in the left or right half plane. To ensure that
the poles will act in a predictable way, then, a minor loop around one of the integrators is used to
move one of the poles onto the real axis in the right half plane. The minor loop is shown in Figure
5-4(b). As shown in Figure 5-3, for positive loop gain the complex pole pair will be in the right
half plane, so the oscillations will grow with each cycle. A limiting circuit is then used to clamp the
magnitude of oscillations to a particular maximum value. As long as this value is not small relative
to the growth in magnitude per cycle, the clamp will not introduce significant distortion.
48
+15
R2
R2
R1
R4
R5
R5
+
R2
speed
R2
−
LM156
R2
R4
C1
speed
−
-15
−
LM156
+
LM156
+
1/4 AD7510
vo1 KR1
C1
R2
−
vo2
R1
R2
−
LM156
+
LM156
+
R2
1/4 AD7510
Figure 5-2: Schematic of voltage control oscillator with amplitude stabilization and direction control.
jω
jω
σ
σ
(a) Positive feedback around one inte- (b) The VCO root locus now has comgrator moves its pole into the right
plex poles in the right half plane.
half plane.
Figure 5-3: Root locus of (a) minor loop and (b) VCO. The minor loop sets a fixed rate of growth
for the oscillations, which are then clipped at a fixed maximum value.
49
The minor loop has the closed-loop transfer function
A
H1 (s) =
A
KR1 C1 s
1−
=
AKR1 C1 s
KR1 C1 s − A
where A is the gain factor from the multiplier (determined by speed input voltage) and K is the
amount of positive feedback. This gives a loop transfer function for the whole system of
L(s) = A
1
R1 R2 C1 C2 s2
L(s) =
AKR1 C1 s
KR1 C1 s − A
=
A2 K
R2 C2 s(KR1 C1 s − A)
AK
R2 C2 s
KR1 C1
s
A
−1
There is one pole at the origin and one at A/KR1 C1 . The closed loop transfer function is
H(s) =
1
AK
RCs( KRC
A s−1)
AK
+ RCs( KRC
A s−1)
H(s) =
=
s2 −
AK
=
− 1) + AK
RCs( KRC
A s
A2
R2 C 2
A2
A
RCK s + R2 C 2
=
The natural frequency of the complex pole pair is ωn =
KR2 C 2 2
s
A
AK
− RCs + AK
ω2
s2 + 2ζωn s + ωn2
A
RC
and the damping ratio is ζ =
−1
2K .
This
means that the growth in oscillation per half-cycle is constant regardless of input voltage.
eζπ = eπ/2K
For a growth of 7.6% per cycle, ζ = 0.025, or K = 20.
In theory, as the multiplier input A goes to zero the system should behave linearly. However,
there is potential for some nonlinearity in the multipliers to bring the pole back out of the right half
plane, resulting in a lower limit in output frequency. It is unclear whether this is in fact the limiting
factor or if other factors in the circuit such as offsets, discussed below, set the lower frequency. One
solution is to add a small amount of A-independent positive feedback in the minor loop so that
positive feedback is always guaranteed.
NB: It is this minor loop that requires the velocity command from the compensator to be broken
up into speed and direction. For a negative A, the sign of the minor loop’s loop transmission would
be inverted.
5.2.2
Direction Control
By switching direction within the VCO itself, the sinusoidal signal can be “unwound,” a process that
allows direction changes to occur without any discontinuities in the motor drive. If the VCO can be
50
vo1
speed
speed
speed
+
−1
R1 C1 s
R
−1
+
dt
−1
KR1 C1 s
− dt
R
−1
R3 C2 s
speed
vo2
(a) Most basic block diagram.
The outputs are taken after
the integrators.
(b) Block diagram with minor loop feedback.
vo1
−1
R1 C1 s
±1
−1
R1 C1 s
±1
vo2
(c) Direction switching
vo1
speed
speed
−1
R1 C1 s
±1
+
−1
+
−1
KR1 C1 s
−1
R3 C2 s
±1
vo2
(d) Complete block diagram
Figure 5-4: Block diagram for VCO, shown at various levels of complexity. The minor loop can
generally be excluded from other analyses because it serves to approximate an ideal system.
51
R
R
vi
−
vo
+
R
1/4 AD7510
Figure 5-5: This circuit has either a gain of +1 or -1 depending on the switch state.
thought of as producing the sine and cosine signals corresponding to travel around the unit circle,
then the direction switching process simply corresponds to changing the direction of this travel. This
can be done by reversing the sign of both integrators. The integrating capacitors hold their state,
so that there is no discontinuity, while the sign reversal undoes the previous integration. By placing
both ±1 gain blocks outside of the minor loop, the minor loop positive feedback can be maintained.
The ±1 circuit block is shown in Figure 5-5. When the switch is open, the voltages at the
inverting and noninverting inputs to the amplifier are the same and are equal to vi since there is
no current into the noninverting terminal node. Thus there must be no current through any of the
resistors and vo = vi . Conversely if the switch is closed, the noninverting terminal is held at ground
and the circuit acts as a standard inverting configuration. The two ±1 circuits are driven by the
direction output of the absolute value circuit in Figure 5-1.
5.2.3
Offset correction
Offsets at the inputs to the op amps and multipliers can cause the VCO signals to saturate and
thus limit the lower range in frequency. For any offset ∆V to exist at the input to the integrating
op amp, there must be a corresponding offset at the input to the multiplier of ∆V /A. When the
multiplication factor A is very small (corresponding to a low frequency of oscillation) this is clearly
a problem because the multiplier input offset will be excessive. Thus the velocity command A has a
lower limit below which the VCO will not oscillate. This situation can be improved by trimming the
offsets in the circuit. The AD524 multiplier includes an offset adjustment pin, so that the output
is equal to (X1 − X2 )(Y1 − Y2 )/10 + Zos . As shown in Figure 5-6, a potentiometer was used to
help trim out the offsets associated with each multiplier/op amp pair. Input current to the op amps
can also affect offsets when it is comparable to the currents charging the integrating capacitors.
At low frequencies, the change in voltage across the capacitor is slow, so the charging current is
correspondingly small. If the op amp input current is high, it will “steal” current from the nominal
charging current. FET-input op amps are used to minimize this effect.
52
+15V
in
Vs
VCO
loop
X1
+
X2
−
R
R
Y1
+
−
X2
−
+
AD532
R
Z
output
Vos
10R
trim
R1
R2
R1
(X1 −X2 )(Y1 −Y2 )
10V
−Vs
-15V
Figure 5-6: Detailed version of the circuits represented as multipliers in Figure 5-2.
53
5.2.4
Results
The circuit was built and tested with the integrator component values R = 31.6kΩ and C = 1µF.
With this configuration, the maximum frequency (corresponding to an input voltage of 12V) is
5.4Hz. The minimum frequency after trimming is 170mHz when the input voltage is 0.4V. For lower
input voltages the circuit does not oscillate. The transition between oscillating and not oscillating
is not continuous – the sine and cosine outputs jump to the saturated value. This discontinuity
will cause the cart to move suddenly. Furthermore, if the loop is operating properly the input to
the VCO will be close to zero, so this condition is unavoidable. The input command voltage was
therefore clamped such that the input cannot drop below this voltage. As a result, the cart is never
stationary, a constraint that does not severely handicap the system.
5.2.5
Three Phase Generation
The sine and cosine generated by the VCO are combined to generate three sinusoids separated in
phase by 120 degrees. The three vectors are:
A = − cos θ
√
1
3
sin θ + cos θ
B=−
2
2
√
1
3
C=
sin θ + cos θ
2
2
The required vectors are generated by three opamp circuits, shown in Figure 5-7.
R1
R1
cos θ
−
+
A
R3
R3
A
√
3R2
−
B
√
3/2R2
R3
A
+
C
−
sin θ
R2
+
B
Figure 5-7: These op amp circuits compute weighted sums of sine and cosine to create three signals
separated by 120◦.
54
5.3
Power Electronics
The three phase signals are used to control the power electronics that drive the motor. Originally, a
linear amplifier topology was attempted. However, the dissipation in the output transistors was too
great to reasonably drive the motor. The relatively low efficiency is especially troublesome because
of the high power levels involved. In order to prevent the transistors from overheating, large heat
sinks and fans are necessary. In order to improve efficiency, then, a switching drive was used. The
design is based on a method described in [10] and first developed by Bose in 1963 [11].
5.3.1
Switching Drive
A single phase of the switching drive is shown in Figure 5-8. This circuit uses a hysteretic control
scheme along with delay circuits that prevent shoot-through. The LM311 comparator, configured
as a Schmitt trigger with positive feedback, makes up the control loop.
Assuming the output of the comparator is high, and that the command input for now is zero, then
the output voltage is +24V, and the 0.047µF capacitor is charging. The load behaves as a low-pass
filter due to its inductance and series resistance. If the driver switching speed is assumed to be
much faster than the corner frequency of the load, the charging can be taken to be roughly linear
as shown in Figure 5-9. When the capacitor voltage reaches the upper limit of the hysteresis band,
the comparator will switch to a low voltage. Now the output of the switching amplifier is -24V, and
the capacitor therefore discharges until it reaches the lower limit of the hysteresis band.
When the command is nonzero, it provides an additional charging current to the capacitor. In
the extreme case, i.e. when the output is driven to the maximum voltage, the capacitor charges at
twice the nominal rate and discharges at a nominally null rate. Since the maximum rate of charge
is only twice the nominal rate, the minimum pulse width of the gate drive is limited to half of
nominal. The frequency of the drive adjusts to match the average output voltage to the command.
The pulse width limit reduces problems with short pulses on the gate drive (incomplete switching,
high switching losses) that can occur in a system where the switching frequency is fixed. Since the
switching drive is for an inherently slow mechanical system, the reduction in switching frequency
is not problematic. Figure 5-9 shows a sketch of the response of the hysteretic control for a step
change in command voltage.
An advantage of this scheme is that the output voltage is always corrected to match the command
within a single switching cycle. The choice of 100kΩ and 0.069µF gives this circuit a maximum
switching frequency of 40kHz when the supplies are ±24V, which is well above the tens of Hertz
frequencies in the rest of the pendulum control loop. Thus the phase shift from this drive will be
ignored in stability analysis.
55
+V
0.1µF
1.1kΩ
IRF5210
TC4427
0.1µF
+15V
MUR120
STPS20L15
+15V
+5V
470Ω
5kΩ
74HC14
SAT
command
+5V
R
1kΩ
−
LM311
0.047µF
+
88.7kΩ
100kΩ
Vc
887Ω
A
SIGNAL GND
B
470Ω
5kΩ
SAT
C
2N2905
-15V
0.1µF
-15V
STPS20L15
IRFI1310
TC4427
MUR120
0.1µF
1.1kΩ
-V
+15V
0.1µF
LM7805
0.1µF
+5V
22µF
15V
Figure 5-8: Switching motor drive, one phase.
56
command
VC
LM311 output
Figure 5-9: Waveforms demonstrating the hysteretic control.
5.3.2
Power Supplies for Switching Drive
Coming up with a reasonably priced power supply capable of supplying the necessary currents is
not trivial. The final design, shown in Figure 5-11, uses four 12V, 7AH lead-acid batteries which
are charged with two 30V wall transformers. Fuses protect the circuitry in case of a short. It is not
recommended to run the motor while the transformers are plugged in as the current draw is too
high and the transformers can fail. The ideal operation is to keep the transformers on a timer and
charge the batteries for a few hours each night and between uses as necessary.
The lead-acid batteries were tested to determine maximum run time of the system. Using a
CBAII battery tester from West Mountain Radio, the battery was discharged at a constant rate of
2.5A until the terminal voltage reached 10V. This discharge rate was chosen based on the roughly
constant 2A current meter reading when a variable power supply was used to power the system.
The results of this test are plotted in Figure 5-10. As the data shows, the system can be run for
about 2.4 hours at a current of 2.5A before the voltage begins to rapidly drop off. This is close to
the nominal value of 7AH. A discrepancy is not unexpected as the amount of energy that can be
extracted from a battery depends on the rate of discharge [12]. Although this section refers to the
supplies as ±24V, therefore, this is only a nominal voltage that depends on the battery state-ofcharge. Since this system is primarily intended to be a lecture demonstration for a feedback course,
this run time is sufficient.
The ±15V supplies are generated using zener diodes. This design was introduced after initial
tests using two separate supplies for ±24 and ±15. It was found that when the supplies where not
applied at exactly the same time, the momentarily high voltage across the TC4427’s would damage
them. Interestingly, another issue with using two sets of power supplies is that many supplies
(for instance, non-synchronous switching supplies or linear regulators) are only equipped to source
current from the positive terminal, not sink it. Because the TC4427 MOSFET drivers are powered
between the +24V and +15V rails, the output current (at peak 1.5A) from these parts is supplied
from the +24V rail into the +15V supply. This current brings up the +15V voltage. The low side
57
Figure 5-10: Terminal voltage of a single battery with constant 2.5A discharge current. The spike
in data at around 4.25AH is due to stopping and restarting the test.
drivers have the same problem, as the -15V rail must supply current to the -24V rail. One remedy
for this problem is to continuously draw a few hundred milliamperes from the ±15V power supplies.
This biases the supplies so that the source currents never go negative. A preferred solution is to
use 5W zener diodes to generate the ±15V supplies, as they can handle currents in this direction
without significant change in voltage. As shown in Figure 5-11, the zener diodes are kept at the
required bias current with 5W, 100Ω resistors. Although zeners are not a particularly low-noise or
(in this simple configuration) solid voltage source, they are practical for this circuit.
Power Circuit Fabrication
A printed circuit board was designed and fabricated for the switching drive. Each phase is built on
a separate PCB to simplify replacement if a phase fails. Since size is not an issue in this application,
critical signal paths and components on the board are spaced apart to avoid coupling. To keep
the boards independent, each one has a separate set of zener diodes to generate the ±15V supplies.
These are located between the high power, switching parts and the logic-level circuits. Separate high
and low power ground planes are also used, with a single connection where the ground connector
meets the board. Smaller copper planes are also used for the high-current traces. The board layout
and a photograph of a populated board are shown in Figure 5-12. In this diagram, the ground and
power planes are represented as outlines only.
58
Figure 5-11: Power supplies for motor drive.
59
120VAC
120VAC
wall transformer, 30VDC/500mA
wall transformer, 30VDC/500mA
20A
12V
12V
20A
20A
12V
20A
12V
0.1µF
0.1µF
22µF
35V
15V
5W
100Ω
5W
22µF, 35V
0.1µF
PWR GND
35V
22µF
100Ω
5W
15V
5W
SIGNAL GND
single connection between grounds
0.1µF
22µF, 35V
-24V
-15V
+15V
+24V
(a) PCB layout
(b) Constructed PCB
Figure 5-12: Each phase of the motor drive is built up on a printed circuit board.
60
Chapter 6
Motor Control
position
feedback
Gs (s)
Gf (s)
VCO
motor/ ẍ
cart
motor
drive
1/s2
x
Ga (s)
θf
fast
pendulum
θs
Glag (s)
slow
pendulum
The primary difficulty in stabilizing even a single pendulum when no feedback is used around
the motor is that there is significant cogging in the mechanical system. Feedback can be used to
reduce this effect. This chapter describes the acceleration loop that was implemented to control the
motor.
6.1
Acceleration Measurement
The signal of interest in this system is acceleration since it directly relates to the angle of the
pendulum, so an acceleration loop is natural. This loop is particularly promising since acceleration
is straightforward to measure. The ADXL213 accelerometer used is well-suited to this application
√
as it has a relatively low full-scale range of ±1.2g and a typical noise floor of 160µg/ Hz [13].
This part has a PWM output for use with microcontrollers; fortunately it also provides analog
outputs at the X_filt and Y_filt pins. A 100nF capacitor on the X_filt pin provides filtering at
1/(2π × 32k × 100nF) = 50Hz.
The ADXL213 is not intended to be used with the X_filt pin as the output, thus the datasheet
does not give a value for the scale factor [Volts]/[g]. This scale factor was measured by comparing
61
the output voltage of the accelerometer when held flat (0g) to the output when vertical (1g). When
flat, X_filt is at 2.62V, and when vertical it is at 3.44V, for a gain of 820mV/g. This scale factor
is not known very precisely since “flat” and “vertical” are determined as best as possible by holding
the board to a standard level.
The printed circuit board built for the accelerometer is shown in Figure 6-1. Because the intent is
not to use the PWM output pins, jumpers are included between these pins and the header connectors.
This prevents high-frequency signals from being routed around the board if unnecessary. Likewise,
a jumper was used to allow the part to enter self-test mode, where the chip output corresponds to
some known value.
(a) PCB layout
(b) Constructed PCB
Figure 6-1: A printed circuit board was built for the accelerometer.
6.2
Characterizing the Plant
An HP 3562A Dynamic Analyzer was used to characterize the plant of the acceleration loop, which
consists of the VCO, motor drive, and motor. The HP 3562A is a dual-channel, fast-Fourier transform based analyzer with frequency capability from DC to 100kHz. By using the dynamic analyzer
to produce Bode plots of the plant, loop compensators could be designed without having to know
and model the exact sources of the plant dynamics.
The plant is measured by placing the A probe at the input to the VCO and the B probe at the
output of the accelerometer as shown in Figure 6-2. For this measurement, the motor is driven openloop. In this configuration the system behaves as a differentiator, so there is no difficulty keeping
the cart on the track, thus no position feedback is necessary. In fact, the dynamic analyzer used has
such a balanced output signal that the cart does not drift on the track during the duration of the
measurement. The system is measured on the dynamic analyzer as shown in Figure 6-3. For this
measurement a gain of 2 is inserted before the VCO to give a full-scale input when combined with the
5V maximum signal out of the dynamic analyzer. Results are shown in Figure 6-3. Since the dynamic
analyzer cannot be left in the paused state for too long (or it seems to assume the measurement
62
is completed), the measurement had to be made in sections. The time between measurements was
also used to recharge the batteries powering the system.
The low-frequency behavior of the plant looks like a differentiator as expected. As frequency
increases, the phase begins to drop off because of higher order poles in the mechanical system. At
around 80Hz, for instance, there is clearly a resonance. The phase shift of the plant crosses −180
degrees at around 40Hz.
6.3
Loop Compensation
Once the plant is well known it is straightforward to compensate the acceleration loop. A crossover
frequency of at least 15Hz is desirable so that the closed-loop system behaves as a double integration
for frequencies at more than a decade above the short pendulum’s natural frequency. Evidently, a
higher crossover frequency is preferred, both for the effect on the loop this block will be a part of
and for increased low-frequency gain in the acceleration loop for disturbance rejection.
In order to close the loop, the sign of the feedback must be determined. This depends on many
factors, such as the order of the motor windings and the physical orientation of the accelerometer.
Thus it was necessary to experimentally determine the sign of the feedback. Since the command
signal and acceleration measurement are added at the summing junction, for negative feedback
they should have opposite signs. This can be tested by first driving the system open-loop with a
constant, positive voltage and observing that the cart accelerates to the right under that condition.
Then the signal is removed and the cart manually accelerated to the right. For negative feedback,
the acceleration signal must go negative.
The loop is compensated with an integrator for high low-frequency gain, followed by a zero a
factor of 4 in frequency below crossover. An additional lead network around crossover ensures plenty
A
source
+
loop opened
VCO
plant
2 Ks
Km
τm s+1
ka s 2
τa +1
accelerometer
B
Figure 6-2: Measurement configuration for results shown in Figure 6-3.
63
(a) Dynamic Analyzer measurement, 300mHz-9Hz, 50 averages, 5V source
(b) Dynamic Analyzer measurement, 7Hz-12Hz, 50 averages, 5V source
Figure 6-3: Measurements of the acceleration loop plant.
64
(c) Dynamic Analyzer measurement, 10Hz-17.8Hz, 20 averages, 2V source
(d) Dynamic Analyzer measurement, 14Hz-20Hz, 20 averages, 2V source
Figure 6-3: Measurements of the acceleration loop plant.
65
(e) Dynamic Analyzer measurement, 16Hz-40Hz, 20 averages, 5V source
(f) Dynamic Analyzer measurement, 38Hz-80Hz, 20 averages, 5V source
Figure 6-3: Measurements of the acceleration loop plant.
66
of phase margin. At crossover, the net phase shift from the integrator plus zero is
φ=6
4
ωc jωc
+1
jωc
!
= tan−1 (4) −
π
= −14◦
2
and the phase from the lead is
φ=6
√
!
10
ωc jωc + 1
√ 1
jωc + 1
10ωc
= 72.5◦ − 17.5◦ = 55◦
so the net phase shift from the compensator is 41 degrees. The schematic of this acceleration loop
is shown in Figure 6-4. A Sallen Key filter is inserted in the feedback path to filter out the coupling
in the accelerometer output due to the coils switching at 20kHz immediately below the sensor. As
derived in [10], when the resistors are chosen to be equal the transfer function of this filter is
1
Vo (s)
= 2 2
Vi (s)
s /ωn + 2ζs/ωn + 1
where
1
ωn = √
R C1 C2
and ζ =
r
C1
C2
In this case, ωn is chosen to be around 1kHz and ζ to be roughly 0.707. With component values
R = 4.99kΩ, C1 = 0.022µF, and C2 = 0.047µF, ωn = 991Hz and ζ = 0.68. The exact component
values are not critical, as some variation in cutoff frequency or even peaking around 1kHz will not
significantly affect the system operation. This allows the use of generic ceramic capacitors, which
may have tolerances of up to ±20%. This filter reduces problems associated with putting a high
frequency signal into a lead network, as well as simply making the signals easier to observe. Since
the cutoff frequency is approximately a factor of 30 above the loop crossover frequency, its phase
contribution can be ignored. The compensator transfer function, excluding the Sallen Key filter, is
1 (R3 C2 s + 1)((R1 + R2 )C1 s + 1)
vo
=
vi
R1 C2
s(R2 C1 s + 1)
A first attempt was made with this compensator to cross over at 20Hz in order to confirm that
the system works as predicted. The component values for this compensator are shown in Table
6.1(a). This system was built and tested. It shows considerable improvement over the open loop
system. Most notably, the visible cogging from the motor is no longer detectable.
With the compensator design confirmed, a more ambitious control loop was attempted. Since the
compensator gives a 41◦ phase bump at the crossover frequency, for a 45◦ phase margin crossover
should be chosen at the frequency at which the plant has just under 180◦ negative phase shift.
According to the measurements in Figure 6-3, this occurs around 38Hz. Component values are
67
+5V
DEMOD
ADXL213
32k
32k
Y-filt
PWM
X Filt
100nF
68
100k
AD524
(×50)
command
offset adjust
+
−
5.11k
22nF
100k
5.11k
R2
47nF
C1
R1
−
+
100k
Figure 6-4: Acceleration loop compensator schematic.
+
−
C2
R3
to VCO
chosen for the compensator as follows. The first zero should be around 1/R3 C2 = ωc /4 = 59.8rps.
Capacitor C2 also sets the gain 1/R1 C2 of the system, which needs to be about 20.8dB based
on the dynamic analyzer measurements and the frequency response of the compensator. The value
C2 = 194nF (nominally 0.22µF) was chosen from the available components. Then R3 must be 86kΩ,
but due to limited availability of resistor values 91kΩ was used instead as the closest value above
nominal. Resistor R1 is set by the gain requirement to be 21.6kΩ. For a pole/zero separation of
α = 10 centered about the crossover frequency, this choice dictates that C1 = 465nF and R2 = 2.7kΩ.
The resulting compensator transfer function is plotted in Figure 6-5 and summarized in Table 6.1(b).
This system was built and tested, showing improvement over the open loop system as with the
20Hz loop. Furthermore, the output of the accelerometer, which looks entirely non-sinusoidal in
the open loop system with a sinusoid input at low frequencies, has a much improved characteristic.
Scope traces are shown in Figure 6-6. For Figure 6-6(a), the loop is broken after the Sallen Key filter.
The circuit is driven with a sine wave at the input to the summing junction and the output is taken
at the output of the AD524. A 1Hz signal in response to the command is just barely discernible.
Figure 6-6(b) an identical input signal is applied to the system with the loop closed. For both cases,
the cart can be observed to move back and forth, but the motion with the loop closed is significantly
smoother.
Table 6.1: Component and frequency values for acceleration loop compensator shown in Figure 6-4.
(a)
(b)
R1
R2
R3
C1
C2
31kΩ
25.4kΩ
3.6kΩ
2.7kΩ
32.4kΩ
91kΩ
680nF (measured) 465nF
1µF (measured) 194nF
ωz1
ωz2
ωp
ωc
4.9Hz
6.8Hz
65.0Hz
20Hz
9.0Hz
11.9Hz
126.8Hz
38Hz
69
103
90
45
|L(jω)|
102
0
101
ωc
100 0
10
10
1
2
10
ω (rps)
6
L
−45
10
−90
3
10
4
(a) Compensator without filter
10
3
90
45
101
0
100
−45
10−1
−90
10−2
−135
|L(jω)|
102
10−3 0
10
6
L
−180
101
102 ωc
103
ω (rps)
104
105
106
(b) Compensator including filter
Figure 6-5: Bode plot of the acceleration loop compensator, with and without the Sallen Key filter.
The Sallen Key filter is at a sufficiently high frequency such that it does not affect the system phase
margin.
70
(a) Open Loop System
(b) Closed Loop System
Figure 6-6: Accelerometer output for the open loop and closed loop systems with a 1Hz sinusoidal
input. The closed loop system is considerably improved.
71
A
source
+
Ga
VCO
plant
(R3 C2 s)((R1 +R2 )C1 s+1)
1
R1 C2 s
(R2 C1 s+1)
K
s
Km
τm s+1
position
ka s 2
τa +1
accelerometer
B
Figure 6-7: Final configuration for the acceleration loop, showing the dynamic analyzer measurement
setup for the results in Figure 6-8.
6.3.1
Closed Loop Operation
The closed loop transfer function of the acceleration loop was measured with the dynamic analyzer
with the configuration shown in 6-7 as shown in Figures 6-8. For these plots, the amplified output
of the accelerometer is taken as the output, and the input is the command signal in Figure 6-4. The
low-frequency behavior of this loop clearly behaves as an ideal double integration as expected. The
upper bandwidth is measured to be approximately 32Hz, somewhat lower than the 40Hz expected.
The phase, however, does not start to drop off until about 6Hz, just above the intended crossover
frequency of the loop of which this block will be a part. Thus this loop accomplishes the goal of not
contributing significant phase shift at all relevant frequencies.
When the accelerometer is driven with a 24V pk-pk (i.e. full-scale) signal, the output of the
accelerometer is 22.6V pk-pk. For an instrumentation amplifier gain of 50, an output of 11.3V from
the accelerometer corresponds to
11.3V
g
1
= 0.275g
50 800mV
Thus the closed loop gain of the acceleration block is 0.024g/V.
72
(a) Dynamic Analyzer measurement, 200mHz-200Hz, 2 averages
(b) Dynamic Analyzer measurement, 2Hz-10Hz, 100 averages
Figure 6-8: Closed loop measurements of the acceleration loop. The loop behavior meets the system
requirements.
73
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74
Chapter 7
Results
Before attempting to stabilize the dual pendulum system, the simpler problem of a single inverted
pendulum was demonstrated. An initial attempt, described in Appendix A, justified the need for a
linear position measurement and a closed-loop motor drive. With these elements included, the fast
pendulum was stabilized as described Section 7.1. Since this single inverted pendulum will form the
minor loop of the dual pendulum system, there are certain requirements for the closed-loop system
which are discussed below. Finally, the dual pendulum system was implemented as described in
Section 7.2.
7.1
Single Inverted Pendulum
The single, fast pendulum was stabilized using the approach proposed in Section 2.2. The system
block diagram is reproduced in Figure 7-1. Initially, the position feedback and lag compensation
were omitted, and only the loop consisting of the lead network, motor, and pendulum was closed.
As described in Section 2.2, the lead network introduces a phase bump that sets the system’s phase
kp
+
motor
Km
s2
G(s)
+
x
+
τ s+1
ατ s+1
−s2 /g
(τp s+1)(τp s−1)
Glag (s)
pendulum
Figure 7-1: Block diagram of single pendulum system with position feedback system and lag compensator.
75
Rpos
R3
position
+
angle
Rang
R1
−
−
input to
motor loop
+
Rf
Ri
C
R2
Figure 7-2: Implementation of summing junction and lead compensator.
jω
−60
−40
−20
jω
−100
σ
−80
−60
−40
−20
σ
(a) Root locus when the lead zero is placed (b) Root locus when the lead zero is placed
below the pendulum poles in frequency.
above the pendulum poles in frequency.
Figure 7-3: Root locus of single pendulum loop for two choices of placement for the lead compensator.
margin.
The circuit implementation of the summing junction and lead network is shown in Figure 7-2.
This circuit has transfer function
R3 (R1 + R2 )Cs + 1
R1
R2 Cs + 1
There are two approaches possible when choosing component values for this compensator. That
is, the lead zero can be placed either below or above the natural frequency of the pendulum. The
root locus for an ideal system using each of these approaches is shown in Figure 7-3. As these
root locus plots show, moving the lead network up in frequency does allow for a higher frequency
crossover, but unless the gain is sufficiently large it results in a closed-loop system with complex
poles. Furthermore, it is important to note that there are additional high frequency poles and
zeros from e.g. the acceleration loop which contribute additional phase shift at higher frequencies.
Because of these elements there is an upper limit to the crossover frequency of this single pendulum
loop. Through experimentation it was determined that this limit corresponds to a loop gain that
76
Table 7.1: Component values for compensator circuit shown in Figure 7-2.
Summing Junction
Lead Network
Rang
Rpos
Ri
Rf
R1
R2
R3
C
27.4kΩ
91kΩ
10kΩ
50kΩ
160kΩ
14.3kΩ
160kΩ
1µF
results in dominant complex poles. The closed-loop transfer function therefore has some peaking in
the frequency domain. This peaking must be avoided when the single pendulum is used as a minor
loop because it significantly lowers the gain margin of the system. Although a higher frequency
lead network should ideally result in a “better” system, then, the slower lead network preferred in
this case due to nonidealities and because this loop will be part of another system. Both types
of compensator were built and tested for this thesis. The results from the higher frequency lead
network compensator are discussed in Appendix B.
The compensator shown in Figure 7-2 was built with the component values as indicated in Table
7.1. This choice of values places the compensator zero at 1.04Hz, just below the pendulum natural
frequency. Figure 7-5 shows the closed loop measurement of the system when the input is an angle
command and the output is the measured pendulum angle as indicated in Figure 7-4. This loop
meets the requirements outlined above: specifically it has no peaking in the frequency domain.
Furthermore, this loop has a relatively high bandwidth, with the magnitude dropping by 3dB at
√
a frequency of about 7Hz. The dual pendulum loop will be ideally closed at ωf ωs = 3.67rps =
0.58Hz. At this frequency, the plot in Figure 7-5(a) shows no measurable negative phase shift. The
step response shown in Figure 7-6 confirms that the closed loop poles are on the real axis.
A
motor
G(s)
Km
s2
x
+
θ
−s2 /g
(τp s+1)(τp s−1)
pendulum
B
Figure 7-4: Measurement configuration for results shown in Figure 7-5.
77
(a) Dynamic Analyzer measurement, 500mHz - 5Hz, 2 averages
(b) Dynamic Analyzer measurement, 4Hz - 10Hz, 8 averages
Figure 7-5: Short pendulum system loop closed loop measurements.
78
Figure 7-6: Step response of the system shown in Figure 7-4. Top trace: measured angle output.
Bottom trace: input angle command.
79
7.1.1
Short Pendulum Position Feedback
Position feedback is not included when the short pendulum is used as the minor loop of a dual
pendulum system. However, for the purposes of lecture demonstrations it makes sense to have a
stand-alone single pendulum system as well as the dual pendulum demonstration. Thus a position
loop was implemented based on the approach in Section 2.2.
In Section 2.2 it was shown that air damping can make additional loop compensation uneccesary
when the position is added as an offset to the angle measurement. With this in mind, a first
attempt at a position control loop was made without lag compensation. In this experiment, the
position feedback constant is chosen as kp = 3.3 degrees/meter. The cart is started in the center of
the track and released. Figure 7-7 shows the measurement of cart position over time. Clearly the
system is unstable.
Figure 7-7: Position measurement output when position feedback is used on the short pendulum
system without a lag compensator. This system is unstable.
A lag network was added to compensate the position feedback system. The complete schematic
of lag, lead, and summing junction is shown in Figure 7-8. The lag network is designed in a noninverting connection so that it can be switched in or out without changing the sign of the loop
transmission. Op amp U1 serves two functions: it adds gain to the lag network to bring the highfrequency gain to unity, and it acts as a buffer to the input of the summing junction to avoid adding
dynamics to the position measurement input. Resistors R4 and R5 set the amount of position
feedback. The position measurement scale is 18.4V/meter, and the angle measurement scale is
1.6V/degree. By adding these two inputs in the ratio Vangle /Vpos = 0.23, the feedback scale factor
80
angle
measurement
position
measurement
or major loop
command
C1
R1
R3
R2
10R3
U1
−
+
81
S3
S2
R6
R4
R5
R7
U2
−
+
Figure 7-8: Complete schematic of compensator implementation for single pendulum loop.
C2
R9
R8
+
U3
−
R10
to motor
drive
Table 7.2: Component values for single pendulum compensator shown in Figure 7-8.
Lag Network
Summing Junction
Lead Network
R1
R2
C1
R3
10R3
R4
R5
R6
R7
R8
R9
C2
R10
2MΩ
511kΩ
2µF
10kΩ
100kΩ
27.4kΩ
91kΩ
10kΩ
50kΩ
160kΩ
14.3kΩ
1µF
160kΩ
kp is determined as:
kp =
[degree]
1.6[Volt]
18.4[Volts]
[meter]
1
= 2.66 degrees/meter
2
Thus when the cart is at one end of the track, the angle is offset by 1.33 degrees. The loop gain
lost by the attenuation of the angle measurement by R5 /(R4 + R5 ) is made up for in the variable
gain block formed by U2, R6 , and R7 . Switches S2 and S3 allow the operator to select whether the
position feedback and lag network are used. More details on operation can be found in Appendix C.
The position feedback was built and tested. When the pendulum cart is manually brought to
one end of the track and released, the system drives the cart back towards the center as expected.
The cart will begin to settle to the location where the offset in angle measurement is corrected by
the offset from the position command to bring the pendulum to true vertical. Due to the minimum
cart speed set by the lower operating limit of the VCO, in addition to the unstable nature of the
system, however, the cart never truly settles. Instead, the cart “hunts” around this location. This
is a characteristic seen in other single pendulum demonstrations, such as the one currently used as
a lecture demonstration in the MIT courses 6.003 and 6.302.
One complication with the position feedback arises when the cart settles to a location offset from
the center of the track. In this case, the integrator producing the position measurement signal is
never reset and will eventually saturate. One solution is to adjust the pendulum offset so that its
sign opposes that of the integrator drift direction. Then, as the integrator drifts it will eventually
push the cart to reset point at the center of the track.
82
7.2
Stabilizing the Dual Inverted Pendulum
Once the single pendulum loop was successfully implemented, the slow pendulum was added for the
dual inverted pendulum system. The minor loop used is identical to the single pendulum system
described in the previous section with position feedback omitted.
As described in Section 2.3, the major loop compensator is a lead network. The zero is placed at
the slow pendulum’s natural frequency, and the pole at the fast pendulum’s natural frequency. This
allows a loop crossover at the geometric
mean of these two frequencies with a maximum theoretical
√
Ls /Lf −1
as derived in Section 2.3.2. The single pendulum closed loop
phase margin of arcsin √
Ls /Lf +1
p
measurements show no measurable phase shift at the ideal crossover frequency fc =
fs ff =
584mHz. This bodes well for system stability.
The compensator implementation is identical to that of the single pendulum compensator. The
topology is shown in Figure 7-2. With the component values R1 = R3 = 144kΩ, R2 = 376kΩ, and
C = 1µF, the lead network has transfer function
G(s) =
0.52s + 1
0.144s + 1
By comparison, the time constants associated with the slow and fast pendulums are 0.52 and 0.143,
respectively.
This system was build and tested before position feedback was attempted. By manually adjusting
the slow pendulum angle offset it was possible to keep the system running on the track. When a
sinusoidal signal was input at the summing junction (in lieu of a position feedback signal), both
the slow and fast pendulum could be observed to move sinusoidally. The slow pendulum could
additionally be disturbed manually and the system would recover. However, these last two tests
are limited by the dynamic range of the fast pendulum. The single pendulum system described in
Section 7.1 can stabilize the fast pendulum for deviations up to six degrees from vertical. Since the
fast pendulum is driven to have an angle from vertical that is twice that of the slow pendulum, this
means that the maximum angle for the slow pendulum is only three degrees from vertical. The short
physical length of the slow pendulum combined with its small dynamic range makes it difficult to
observe the small-signal response. For a lecture demonstration, the physical length will have to be
increased.
7.2.1
Position Feedback
Although not explicitly part of the problem statement, position feedback is necessary for any reasonable demonstration as the cart will run off the track if it is omitted. The approach for closing a
position feedback loop for the dual system is similar to that used for the single inverted pendulum.
83
Figure 7-9: Position measurement output when position feedback is connected around the dual
position system with no additional compensation.
Table 7.3: Component values for dual pendulum compensator shown in Figure 7-10.
Lag Network
Summing Junction
Lead Network
R1
R2
C1
R3
R4
R5
R6
R7
R8
R9
C2
72kΩ
75kΩ
10µF
294kΩ
22.6kΩ
111.3kΩ
50kΩ
376kΩ
511kΩ
376kΩ
1µF
In order to measure the lower crossover frequency of the major loop, the position is first connected
with no addition loop compensation. When R5 = 111.3kΩ, the cart oscillates at 96mHz as shown in
Figure 7-9. Thus the lag compensator should be centered about this frequency. In order to limit the
lower frequency of the pole from the lag network, the separation between the lag pole and zero is
chosen to be only a factor of 5 in frequency. This value was chosen to avoid a very slow rise time in
response to a step in angle. Otherwise, the process of manually trimming out the pendulum angle
as described in Appendix C becomes tedious if the system takes over 30 seconds to settle.
A schematic of the dual pendulum loop is shown in Figure 7-10, with component values given in
Table 7.3. This circuit was built and tested, and seems to behave as expected with both pendulums
held upright, although the system response has not yet been thoroughly measured.
84
Figure 7-10: Circuit implementation of compensator for dual pendulum loop.
85
angle
measurement
R1
+
−
R2
R3
C1
R4
R5
+
−
R6
position
measurement
C2
R7
R8
+
−
R9
input to
minor loop
7.3
Conclusions
This thesis has demonstrated that it is possible to implement a dual inverted pendulum system using
a stable minor loop approach. Furthermore, the systems built along the way, such as the motor drive
and acceleration loop, provide a strong foundation for possible improvement to the system.
One of the major successes of this project was the use of counterbalancing to produce a pendulum with a very slow time constant. This technique allows for a high phase margin in the dual
pendulum system, making demonstrations of stability more impressive, while avoiding a physically
large structure. Before being presented as a lecture demonstration, however, the upwards-facing
pendulum part should be extended so that it is physically longer than the fast pendulum. It seems
that this aesthetic modification, although it will not affect the stability of the system, will make the
system more immediately understandable.
There are some additional adjustments that should be made before the system is robust enough
to be used as a lecture demonstration. For instance, one practical improvement would be to reduce
the power consumption of the circuits located on the cart to extend its run time, since for this thesis
no attempt was made to reduce power consumption. However, such improvements are related to the
reliability of the system over repeated demonstrations rather than the system stability.
The single inverted pendulum system, which was initially intended to serve only as a minor loop,
can be used as a demonstration of the single pendulum feedback system. This system is very robust
and has very smooth operation, making it ideal for a lecture demonstration. Because this system
is configurable, it can additionally be used to demonstrate the importance of the position feedback
compensation. This ability will allow for a more complete discussion of the non-idealities involved,
rather than a system that “just works.”
This thesis attempts to convey the complexity of compensating what appears to be a straightforward system in the real world. It is interesting to note that the majority of the time spent on
this thesis was focused first on the machining of the mechanical system, and then on the design and
implementation of all of the necessary circuits that do not appear in the theoretical analysis, such
as the motor drive, or the implementation of the pendulum angle and cart position measurements.
Only in the final months were the pendulums stabilized, and in fact this process (particularly the
single pendulum system), was relatively straightforward.
86
Appendix A
Temporary Single Inverted
Pendulum
The first pendulum stabilized for this thesis was a single pendulum with effective length 90cm. This
attempt was made in order to make sure that the mechanical system and supporting electronics
such as the motor drive were sufficient to stabilize the faster pendulum that was eventually used
as the minor loop pendulum in the dual inverted pendulum system. This experiment is interesting
primarily because it demonstrated a need for both a closed-loop motor drive and a linear position
measurement.
This temporary single pendulum system was built before an acceleration loop was closed around
the motor drive. The open-loop motor drive affects the system significantly. First, it introduces
a low-frequency pole measured to be around 13 Hz that contributes negative phase shift to the
transfer function, limiting the phase margin. Furthermore, since the stator windings are not perfectly
sinusoidal, the cart does not in fact move at a constant velocity when the velocity command to the
open loop system is constant. Instead, it accelerates at each pole winding, resulting in the sort of
behavior shown in Figure 6-6(a). When a feedback loop is used, these spikes in acceleration are
diminished as shown in Figure 6-6(b). Finally, it is possible when using the open-loop drive to
command the magnetic wave to accelerate at a higher rate than the inertia of the cart allows the
rotor to match. When the acceleration is too great, the motor will operate asynchronously. This
problem arises because the open-loop drive has a velocity command input, making the slope of this
input effectively an acceleration command. By limiting the rate of change of the velocity command,
the acceleration is limited to a value that can be met by the cart. The slew rate limiter schematic is
shown in Figure A-1. This circuit is based on the four-diode gate, with the charging rate of capacitor
C limited by the bias current Ic . The rate of change of the voltage across the capacitor is limited
87
+15V
+15V
200Ω
200Ω
Ic
34.5kΩ
velocity
command
+
100kΩ
C
4.7µF
200Ω
-15V
−
input
to VCO
200Ω
-15V
Figure A-1: An adjustable slew rate limiter was used to limit the acceleration to avoid slip when
the motor was driven open loop.
to dV/dt = Ic /C.
In order to compensate this system, the approach derived in Section 2.2 must be slightly modified
to account for the change in the motor transfer function from Km /s2 to Km /s(τm s + 1). Because
there is only one integration from the motor, the compensator must include an integration as shown
in the block diagram in Figure A-2. The requirement of an additional integration makes it possible to
use a compensator without the higher frequency pole included in the analysis in Section 2.2. A pole
is included in the original compensator in order to limit its high-frequency gain, effectively avoiding
building an analog differentiator. If a pole is included at the origin, the gain will be constant above
the lead zero and there is no need for an additional pole. The transfer function of this circuit is
vo
RCs + 1
=
vi
2RCs
The circuit topology for this compensator is shown in Figure A-3.
To keep the cart on the track, the basic position measurement described in 4.2.1 was used. This
position measurement scheme divides the motor track into eight discrete segments, each with an
associated value. The values are added as an offset to the angle measurement to drive the cart to
the center. As mentioned in Section 4.2.1, a low-pass filter was used to limit the rate of change
88
kp
+
+
G(s)
motor
K(τ s+1)
s
Km
s(τm s+1)
x
+
θ
−s2 /g
(τp s+1)(τp s−1)
pendulum
Figure A-2: When the motor is driven open loop, an additional integration must be added in the
single pendulum loop compensator.
R1
angle
+
position
R1
R
to VCO
−
C
Figure A-3: The single inverted pendulum compensator consists of an integrator and a zero around
the natural frequency of the pendulum.
89
of the position offset. Avoiding a step input in this way is beneficial for a system with low phase
margin, since any overshoot might bring the pendulum angle outside of the operating range of the
system.
The temporary pendulum system demonstrated a need for two systems that were ultimately
implemented for this system: a linear position measurement and a closed loop motor drive. Both of
these additions reduce the disturbances in the pendulum system. The analog approach to position
measurement satisfies the need for a position signal that does not have step changes. Similarly, the
acceleration loop removes the unwanted accelerations due to motor saliency. The 90cm pendulum
system was able to reject both of these disturbance sources, but the dual pendulum system inherently
has lower stability margins, so it was necessary to remove these disturbances from the system.
90
Appendix B
An Alternative Single Pendulum
Loop
As desribed in Section 7.1, there are two approaches to stabilizing the single inverted pendulum
depending on the placement of the lead network in frequency. Based on the theoretical model of
the system, the lead network should be able to be placed at an arbitrarily high frequency, with the
loop gain increased accordingly. Thus the initial approach to stabilize the single inverted pendulum
was to incrementally increase the loop gain and lead frequency in an attempt to achieve the highest
possible crossover frequency. The results below refer to a single inverted pendulum without position
feedback.
The compensator implementation is reproduced in Figure B-1. This circuit is identical in topology
to that used in the final version of the short pendulum system as described in Section 7.1. The
circuit was first built and tested with the component values shown in Table B.1(a). This choice of
components places the compensator zero at the frequency of the short pendulum pole. By adjusting
the variable gain block, the system can be made to keep the pendulum upright, although the cart
of course has the tendency to drift off the track. The system can run continuously for some time if
the pendulum is tapped by hand toward the center of the track when it approaches the ends. This
small interference with the system does not significantly affect its performance.
The lead network frequency was incrementally increased with the values shown in Table B.1(b)(e). Note that in this table, the gain value Gvar does not directly correspond to the loop gain due
to other scale factors associated with the angle measurement and motor drive. In theory, the gain
should increase by k 2 for every k increase in lead frequency. However, Table B.1 shows that this is
only true for low frequencies. At higher frequencies there is apparently negative phase shift from
the non-idealities in the system, presumably from the 40Hz motor drive, so the loop gain cannot
91
Rpos
R3
position
+
angle
Rang
R1
−
−
input to
motor loop
+
Ri
Rf
C
R2
Figure B-1: Implementation of summing junction and lead compensator.
be increased without causing instability. Naturally the maximum phase margin is also limited at
higher frequencies, as demonstrated by the increasing peak overshoot (Po ) in response to a step
input. Of these attempts, system (d) gives the best results since it has a combination of high
crossover frequency and damping ratio.
The frequency response was tested with the dynamic analyzer for component choice (d). The
results of this measurement are shown in Figure B-2. The peaking of 4.3dB at 4.9Hz corroborates
the damping ratio derived from the small signal response, shown in Figure B-3. When the lead zero
and loop gain were increased further, the system could not be stabilized.
Based on these results, it seems that the crossover frequency is limited to such an extent that
the frequency domain peaking that results from this approach outweigh any potential benefits of a
higher frequency lead. The limit is probably due to phase shift from additional poles that are not
included in this analysis. In addition, when the loop gain is large it becomes difficult to stabilize
the system due to saturating signals. This approach was therefore discarded in favor of the one
described in Section 7.1, since minor loop peaking in the frequency domain cannot be tolerated in
the dual pendulum system.
Table B.1: Various iterations of component values and gain and peak overshoot measurements for
the compensator shown in Figure B-1. The small signal measurements for system (d) are shown in
Figure B-2 and Figure B-3.
R1
R2
R3
C
max Gvar
measured Po
(a)
(b)
(c)
(d)
(e)
28kΩ
2.2kΩ
28kΩ
4.7µF
0.51
28kΩ
2.2kΩ
28kΩ
2µF
3.9
28kΩ
2.2kΩ
28kΩ
660nF
9.79
28kΩ
2.2kΩ
28kΩ
660nF
5.2
1.29
(ζ ≈ 0.4)
28kΩ
2.2kΩ
28kΩ
330nF
5.3
1.55
(ζ ≈ 0.2)
92
Figure B-2: The single short pendulum system: closed loop measurement from dynamic analyzer.
93
(a) Step Response
(b) Sinusoidal Response
Figure B-3: Small signal response of short pendulum system without position feedback. In both
figures, the top trace is the output of the angle sensor (taken after the Sallen-Key filter) and the
bottom trace is the command signal. Note the large amplitude of the angle signal in (b), which
corresponds to a deviation from vertical of up to 4 degrees.
94
Appendix C
Operating Instructions
This thesis is set up to perform a variety of demonstrations as well as the dual inverted pendulum
system. Each demonstration requires some initial setup as well as offset adjustments. This appendix
describes the configuration and operation of the demonstrations.
C.1
Configuration
The cart can be configured for three types of demonstrations: the dual inverted pendulum system,
and the single inverted pendulum with and without position feedback. Three switches control the
configuration as summarized in Table C.1. Figure C-1 shows a sketch of the physical layout of the
cart circuit board with the approximate locations of the switches.
a
S3
a
b
a
POT4
S2
S1
b
b
POT3
POT2
POT1
Figure C-1: Sketch of physical locations of switches on cart board.
The angle measurement offsets and pendulum loop gains are adjusted with the four potentiometers sketched in Figure C-1. The fast and slow pendulum angle measurements are offset with
95
Table C.1: Table of switch settings for various demonstrations.
S1
S2
S3
Demonstration
a
a
a
Single pendulum - no position feedback
a
b
a
Single pendulum - uncompensated pos. feedback
a
b
b
Single pendulum - compensated pos. feedback
b
b
a
Dual pendulum system with position feedback
potentiometers POT1 and POT2, respectively. The minor (single) pendulum loop gain is adjusted
with POT4, and the major loop gain with POT3.
C.2
Startup
This thesis is designed for small-signal operation around vertical and ignores the problem of getting
the pendulum upright in the first place. A manual approach to startup is described below.
The single pendulum system has a double integration because of the acceleration loop, so the
signals will saturate if the loop is not closed. To start the system, then, the signals must be coaxed
into the operating range for the feedback to kick in. Startup is best achieved by bringing the cart
to one side of the track, then leaning the pendulum to the opposite side and allowing the cart to
“catch” the pendulum from one side. The pendulum loop is too fast to simply hold the pendulum
at small angles by tracking the movement of the cart, a technique that works for the slower (1
meter) pendulum. Another effective technique is to slowly increase the loop gain while holding the
pendulum upright until the system can be felt taking control.
The angle offset seems to be slightly different each time the system is started up, probably due
to the impact from the pendulum falling when the system is shut off. There is a potentiometer on
the pendulum cart to adjust this offset (see Section C.1 above). When position feedback is being
used this should be adjusted with the cart as close to the center as possible, ideally with the position
reset brush contacting the copper foil pad. The angle offset should be corrected before adding an
input or the second pendulum.
The general approach to the dual pendulum system is to get the fast pendulum upright first
by commanding it to an angle of zero, then switching in the slow pendulum. The slow pendulum
is held upright while the minor loop input is observed on an oscilloscope. By adjusting the angle
offset, the minor loop input is brought approximately to zero, then the system is switched into the
dual pendulum mode. The two pendulums should be trimmed as closely as possible.
96
Appendix D
List of Symbols
A
A, B
fs , ff
g
Glag
G(s)
Gs (s), Gf (s)
Gvar
kℓ
Km
km
kp
kρ
ℓ
ℓs , ℓf
l
L
L(s)
Po
x
ζ
θp
θs , θf
θz
µ
τc
τg
φM
ωc
ωs , ωf
VCO multiplier gain
dynamic analyzer inputs, output is B/A(s)
slow, fast pendulum natural frequency in Hertz
gravitational constant, 9.81m/s2
lag compensator
general loop compensator
slow, fast pendulum loop compensator
gain associated with variable gain block
ratio of lengths in slow pendulum assembly
motor block gain constant
ratio of masses in slow pendulum assembly
position feedback constant
ratio of linear density of steel cylinder to PETG hollow rod
pendulum effective length
slow, fast pendulum effective length
length integration variable
physical length of non-ideal pendulum.
system loop transfer function
peak overshoot in system step response
cart position
system damping ratio
angle contributed by a zero in the transfer function
slow, fast pendulum angle from vertical
angle contributed by a pole in the transfer function
linear density of short pendulum
torque acting on pendulum due to acceleration of the cart
torque acting on pendulum due to gravity
system phase margin
system crossover frequency in radians per second
slow, fast pendulum natural frequency in radians per second
97
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98
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